2
.
The cyclical component of the series is deﬁned as y
c
t
= y
t
−g
t
. The parameter
µ controls the smoothness of the trend component. For quarterly data, µ is
typically recorded as 1600. Taking the derivative with respect to g
t
yields
∞

t =−∞

µg
t +2
− 4µg
t +1
+ (1 + 6µ)g
t
− 4µg
t −1
+µg
t −2

=
∞

t =−∞
y
t
.
(1.1)
Using the lag operator notation and simplifying, the cyclical component can
be represented as
y
c
t
= y
t
− g
t
=
µ(1 − L
−1
)
2
(1 − L)
2
1 +µ(1 − L
−1
)
2
(1 − L)
2
y
t
. (1.2)
In practice, the ﬁlter is applied using a ﬁnite sample. The general equation
characterizing the ﬁlter, Eq. (1.1), is modiﬁed at the beginning and end
of the sample. The properties of the HP ﬁlter have been studied by many
authors, including Singleton [192], King and Rebelo [130], and Cogley and
Nason [69]. Cogley and Nason have argued, in particular, that business
cycle dynamics obtained by using a HP de-trending procedure depend on
the properties of the underlying data. If these are trend-stationary, then the
de-trending procedure has favorable properties; if the underlying data are
October 9, 2009 15:12 9in x 6in b808-ch02
14 Business Cycles: Fact, Fallacy and Fantasy
difference-stationary, however, application of the HP ﬁlter induces spurious
business cycle ﬂuctuations.
Asecond approach is based on the spectral analysis of economic time series.
The band-pass ﬁlter developed by Baxter and King [37] “ﬁlters” out both the
long-run trend and the high-frequency movements in a given time series while
retaining those components associated with periodicities of typical business
cycle durations, namely, periodicities between six quarters and eight years. The
band-pass ﬁlter of Baxter and King [37] is obtained by applying a K th-order
moving average to a given time series:
y
∗
t
=
K

k=K
a
k
y
t −k
, (1.3)
where the moving average coefﬁcients are chosen to be symmetric, a
k
= a
−k
for k = 1, . . . , K . They showthat if the sumof the moving average coefﬁcients
is zero,

K
k=K
a
k
= 0, then it has trend elimination properties. In particular,
they show that the lag polynomial describing the K th-order moving average
can be written as
a(L) = (1 − L)(1 − L
−1
)ψ(L), (1.4)
where ψ(L) is a (K −1)-order symmetric moving average polynomial. Hence,
the Baxter–King ﬁlter will eliminate deterministic quadratic trends or render
stationary series that are integrated up to order two, i.e, I (2) or less. The ﬁlter
is designed to have a number of other properties, including the property that
the results should not depend on the sample size and that it does not alter the
timing relations between series at any frequency. Baxter and King derive the
band-pass ﬁlter by considering low-pass and high-pass ﬁlters with the required
properties. Let s denote the number of observations in a year. The band-pass
ﬁlter retains the movements in a series for the frequencies associated with
the periodicities of 0.5s and 8s. The frequency response function of the ideal
band-pass ﬁlter is deﬁned as
β
bp
(ω) = I (2π/(8s) ≤ ω ≤ 2π/(0.5s)),
where I (·) is the indicator function. It turns out that the time-domain
representation of the band-pass ﬁlter is an inﬁnite moving average. However,
once the ideal ﬁlter’s weights are found in the time domain, the optimal
October 9, 2009 15:12 9in x 6in b808-ch02
Facts 15
7.2
7.6
8.0
8.4
8.8
9.2
9.6
50 55 60 65 70 75 80 85 90 95 00 05
BPTEST USA
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
50 55 60 65 70 75 80 85 90 95 00 05
BPCYCLEUSA
Fig. 2.2. Trend and Cyclical Components in USA GDP.
approximating ﬁlter with maximum lag K is obtained by truncating the
ideal ﬁlter’s weights at lag K . Baxter and King [37] employ a ﬁnite-order
approximation to obtain the coefﬁcients of this ﬁlter with a truncation of
K = 3 years or K = 12 quarters. In what follows, we use this approach to
generate the cyclical components of the different series.
Figure 2.2 illustrates the logarithm of real GDP for the US and its trend
and components estimated according to the Baxter–King ﬁlter for the period
1947(I)–2005(III). We observe that real GDP displays a marked positive
trend over the sample period, showing the large gains in productivity and
growth attained over this period. We also observe the cyclical troughs and
peaks of economic activity associated with the business cycle. We can observe
the cyclical downturns and upturns corresponding to the NBERbusiness cycle
reference dates from this graph. The cyclical component tracks very well the
US business cycle as identiﬁed by the NBER. Nevertheless, the notion of a
“business cycle” is also concerned with the co-movement of a large number of
economic variables. Section 2.2 discusses these properties of business cycles.
2.2. STYLIZED FACTS
In this section, we provide some stylized facts of business cycles. These facts
constitute important benchmarks by which to judge the performance of
alternative models of business cycles. We consider a measure of the business
cycle as the co-movement of the cyclical behavior of individual series with
October 9, 2009 15:12 9in x 6in b808-ch02
16 Business Cycles: Fact, Fallacy and Fantasy
the cyclical component of real output. According to this approach, variables
that move in the same direction over the cycle as real output are procyclical.
Variables that move in the opposite direction (rise during recessions and
fall in expansions) are countercyclical. Variables that display little correlation
with output over the cycle are called acyclical. We can also examine whether
different time series are out of phase with real GDP. For example, a leading
indicator reaches a peak before real GDP reaches its peak and bottoms out
(reaches a trough) before real GDP. Leading indicators are useful for predicting
subsequent changes inreal GDP. Coincident indicators reacha peak or a trough
at roughly the same time as real GDP. Finally, lagging indicators reach a peak
or trough after real GDP.
The stylized facts of business cycles are typically deﬁned in terms of the
behavior of the main components of GDP, hours, productivity, real wages,
asset returns and prices, and monetary aggregates. These have been described
in a number of economic studies.
1
Information on leading indicators is also
collected by the Conference Board.
2
The salient facts of a business cycle can be described as follows:
1. Real output across virtually all sectors of the economy moves together.
In other words, the contemporaneous correlation of output in different
sectors of the economy is large and positive. Exceptions are production
of agricultural goods and natural resources, which are not especially
procyclical.
2. Consumption, investment, inventories, and imports are all strongly
procyclical. Consumption of durables is much more volatile than
consumption of nondurable goods and services. Consumption of
1
Stock and Watson [197] use data on 71 variables to characterize US business cycle phenomena over
the period 1953–1996. They make use of the Baxter–King band-pass ﬁlter to identify the trend versus
cyclical components of each series. Among other measures, they consider the co-movement of the cyclical
component of GDP with the cyclical component of each series, and the cross-correlations of the cyclical
component of GDP with the cyclical component of each series for lags and leads up to six periods. Backus
and Kehoe [23] analyze the properties of historical business cycles for ten developed countries using a
century-long dataset up to the 1980s. They use the HP ﬁlter to derive the cyclical components of the
different series, and separate their sample period into the pre-World War I era, the interwar era between
World War I and II, and the post-World War II era.
2
Early work on developing a composite index of leading indicators is due to Mitchell and Burns [164].
Such a composite index was made ofﬁcial by the US Department of Commerce in 1968, and passed over
to the Conference Board in 1995.
October 9, 2009 15:12 9in x 6in b808-ch02
Facts 17
durable goods ﬂuctuates more than GDP, whereas nondurables ﬂuctuate
considerably less.
3. Investment in equipment and nonresidential structures is procyclical with
a lag. Investment in residential structures is procyclical and highly volatile.
4. Government spending tends to be acyclical. The correlation between
government expenditures and output is nearly zero.
5. Net exports are countercyclical. The correlation with output is generally
negative, but weakly so. Since imports are more strongly procyclical than
exports, the trade balance tends to be countercyclical.
6. Total employment, employee hours, and capacity utilization are all
strongly procyclical. The employment series lags the business cycle by
a quarter, while capacity utilization tends to be coincident.
7. Employment ﬂuctuates almost as much as output and total hours of
work, while average weekly hours ﬂuctuate much less. The implication
is that most ﬂuctuations in total hours result from movements in
and out of the work force rather than adjustments in average hours
of work.
8. Real wages are procyclical or acyclical. They have not displayed a steady
pattern in terms of variability to GDP or in terms of leading, coincident,
or lagging indicators.
9. Productivity is slightly procyclical, but both real wages and productivity
vary considerably less than output.
10. Proﬁts are highly volatile.
11. Nominal interest rates tend to be procyclical. The yield curve which shows
the rates of return on bonds of different maturities tends to be upward-
sloping during an expansion and downward-sloping at the onset of a
recession. That is, an expansion is characterized by expectations of higher
interest rates at longer horizons whereas a recession typically signals a
decline in long-term interest rates relative to short-term rates, namely, an
inverted yield curve.
12. Velocity and the money supply are procyclical.
13. The risk premium for holding private debt, or the yield spread between
corporate paper and Treasury bills with six months’ maturity, tends to
shrink during expansions and increase during recessions. The reason for
this countercyclical behavior is likely to be changes in default risk.
October 9, 2009 15:12 9in x 6in b808-ch02
18 Business Cycles: Fact, Fallacy and Fantasy
14. The stock market is positively related to the subsequent growth rate of real
GDP. In this sense, changes in stock prices have been taken as providing
information about the future course of the real economy. Between 1945
and 1980, the stock market fell in the quarter before each of the eight
recessions, although it is important to emphasize that the market has fallen
without a subsequent recession.
15. Money (M2) is procyclical and tends to be a leading indicator of output.
However, the procyclicality of M2 has diminished since the 1980s.
16. The behavior of prices and inﬂation appears to have changed over time.
In the pre-WWI period and interwar period, inﬂation was procyclical
with a very low mean. Since the early 1980s, inﬂation appears to
be countercyclical. A similar change appears to have characterized the
behavior of price-level ﬂuctuations.
17. The standard deviation of inﬂation is lower than that of real GDP.
18. Inﬂation is a coincident indicator.
19. There is also a marked increase in the persistence of inﬂation after WWII.
20. Finally, contemporaneous correlations between output ﬂuctuations in
different countries were highest in the interwar period, reﬂecting the
common experience of the Great Depression, with the exception of
Germany and Japan. The correlation is typically larger in the post-war
period than in the pre-war period.
This set of facts constitutes a benchmark which a successful business
cycle model should meet. The ﬁndings concerning the behavior of hours,
employment, real wages, and productivity have proved among the most
difﬁcult to reconcile by current business cycle theories. The changes in the
cyclical behavior of prices and inﬂation also have implications for the so-
called impulses and propagation mechanisms of business cycles. In the post-
WWII era, there is more evidence of supply-side-driven ﬂuctuations in output.
Witness the impact of the large oil shocks in the 1970s, a topic to which we
will return; hence, the increased persistence of inﬂation and the countercyclical
behavior of prices observed in this era. By contrast, prices and inﬂation were
procyclical in the pre-WWII era. One way to rationalize this phenomenon
may be in terms of monetary policy that is more accommodative — under a
gold standard, for example. Other papers have generated business cycle facts
using data over long samples. Chadha and Nolan [62] identify the stylized facts
October 9, 2009 15:12 9in x 6in b808-ch02
Facts 19
of business cycles for the UK economy over a long sample period. A’Hearna
and Woitek [2] establish the facts of business cycles in the late 19th century
using spectral techniques.
The above ﬁndings can also shed light on whether output ﬂuctuations
have moderated in the post-WWII era. On the one hand, Christina Romer
[180, 181] has argued that the ﬁnding of lower output variability in the post-
WWII period is due to changes in the measurement of output in the pre-
and post-WWII eras. If the higher-quality data in the post-WWII period are
subjected to a transformation that makes the pre- and post-WWII data of
comparable quality, then one discerns no change in output ﬂuctuations across
the two periods. While her ﬁndings have garnered much interest, Backus and
Kehoe [23] argue that it is difﬁcult to reach a ﬁrm conclusion on this point
based on US data alone. They suggest that if one considers additional evidence
based on a larger sample of countries over the different periods, the variability
of output appears to have diminished in the post-WWII period. Stock and
Watson [198] seek to avoid the problemof poor data quality in the pre-WWII
period, and compare the volatility of 21 different variables in the 20 years since
the 1980s andthe 40years inthe periodfollowingWWII upto the 1980s. They
ﬁnd that the standard deviations of a typical variable in their sample declined
by about 20–40% since the 1980s. They attribute around 20–30% of the
reduction in output volatility to better monetary policy, 15%to smaller shocks
to productivity, and another 15% to reduced shocks to food and commodity
prices. Their ﬁndings also provide evidence on the factors behind the “Great
Moderation”.
2.3. THE EURO AREA BUSINESS CYCLE
Much of the early work on the European business cycle was concerned with
examining the impact of monetary union arrangements on economic activity.
Artis andZhang [18, 19] investigate the relationshipof the EuropeanExchange
Rate Mechanism (ERM) to the international business cycle in terms of the
linkage and synchronization of cyclical ﬂuctuations between countries. Their
ﬁndings suggest the emergence of a group-speciﬁc European business cycle
since the formation of the ERM, which is independent of the US cycle. Artis,
Kontolemis, and Osborn [20] propose business cycle turning points for a
number of countries based on industrial production. The countries selected are
October 9, 2009 15:12 9in x 6in b808-ch02
20 Business Cycles: Fact, Fallacy and Fantasy
the G7 countries along with the prominent European countries. They use this
information to examine the international nature of cyclical movements, and to
determine whether cyclical movements are similar across different countries.
They also consider the lead/lag relationships between countries at peaks and
troughs.
Artis, Marcellino, and Proietti [21] discuss alternative approaches to dating
euro area business cycles. As in the Stock and Watson [197] approach, they
distinguish between classical and growth cycles. The euro area experience
makes consideration of both types of business cycles relevant. Whereas in
the post-WWII era the euro area countries exhibited high rates of growth,
making growth cycles the relevant concept, in recent years growth has slowed
and even absolute declines in real GDP have been observed, implying that
classical cycles should also be considered. These authors also articulate a
formal model of turning points based on restrictions imposed on a Markov
process.
3
To describe this algorithm, suppose that the economy can be in one
of two mutually exclusive states, expansion E
t
or recession R
t
. Suppose that
a peak terminates an expansion, and a trough terminates a recession. The
Markov process distinguishes between turning points within the two states by
assuming that
E
t
=

EC
t
expansion continuation
P
t
peak
,
R
t
=

RC
t
recession continuation
T
t
trough
.
Let p
EP
= Pr(P
t +1
|EC
t
) denote the probability of transiting to a peak,
conditional on being in an expansion, which implies that the probability of
continuing an expansion p
EE
= Pr(EC
t +1
|EC
t
) is given by p
EE
= 1 −
p
EP
. Likewise, deﬁne p
RT
= Pr(T
t +1
|RC
t
) as the probability of transiting
to a trough, conditional on being in a contraction. Then, the probability of
continuing a contraction p
RR
= Pr(RC
t +1
|RC
t
) is p
RR
= 1 − p
RT
. Let S
t
denote a discrete random variable that follows a ﬁrst-order Markov process
3
This algorithm follows Harding and Pagan [115], who extended the Bry and Boschan [49] algorithm
to a quarterly setting.
October 9, 2009 15:12 9in x 6in b808-ch02
Facts 21
with four states and transition probability matrix as follows:
EC
t +1
P
t +1
RC
t +1
T
t +1
EC
t
p
EE
p
EP
0 0
P
t
0 0 1 0
RC
t
0 0 p
RR
p
RT
T
t
1 0 0 0
The dating rules impose minimum durations on a phase, expansion or
contraction, and on complete cycles, peak-to-peak or trough-to-trough. The
duration of a phase is required to be two quarters, which is automatically
satisﬁed since the states (EC
t
, P
t
) both belong to an expansion and (RC
t
, T
t
)
belong to a contraction. The minimum duration of a complete cycle is given
as ﬁve quarters. Hence, a peak at date t − 4, P
t −4
, can only be followed by a
peak at t + 1, P
t +1
. A similar condition exists for troughs.
Now consider applying this algorithm to dating classical cycles. Let y
t
denote the underlying series. Then, we can deﬁne an expansion termination
sequence, ETS
t
, and a recession termination sequence, RTS
t
, respectively, as
follows:
ETS
t
= {(y
t +1
< 0) ∪ (
2
y
t +2
< 0)}
RTS
t
= {(y
t +1
> 0) ∪ (
2
y
t +2
> 0)}.
Thus, y
t +1
= y
t +1
− y
t
< 0 and
2
y
t +2
= y
t +2
− y
t
< 0 represent
the candidate sequence for terminating an expansion, which deﬁnes a peak.
Likewise, y
t +1
= y
t +1
− y
t
> 0 and
2
y
t +2
= y
t +2
− y
t
> 0 represent
the candidate sequence for terminating a contraction, which deﬁnes a trough.
The algorithm is completed by ﬁnding the probability that the economy will
transit to a peak, conditional on having been in an expansionary phase, p
EP
.
This is the joint event that the history at time t , (S
t −4
, S
t −3
, S
t −2
, S
t −1
, S
t
),
features an expansionary state at time t , S
t
= EC
t
, and that ETS
t
is true.
Else, if ETS
t
is false, the expansion continues. Likewise, the probability
that the economy will transit to a trough, conditional on having been in a
contractionary phase, is p
RT
. This is the joint event that the history at time t ,
(S
t −4
, S
t −3
, S
t −2
, S
t −1
, S
t
), features a contractionary state at time t , S
t
= RC
t
,
October 9, 2009 15:12 9in x 6in b808-ch02
22 Business Cycles: Fact, Fallacy and Fantasy
and that RTS
t
is true. Otherwise, if RTS
t
is false, the contraction continues.
The dating of deviation cycles is achieved by modifying this algorithm to
account for the fact that a peak cannot occur if output is below trend, and a
trough cannot occur if output is above it.
Artis et al. [21] also examine the properties of alternative ﬁlters such as the
Hodrick–Prescott and Baxter–King ﬁlters for decomposing a time series into
trend and cyclical components. They implement their procedures using data
on euro area GDP and its components for the European Central Bank’s (ECB)
Area-Wide Model for the period 1970–2001.
4
Table 2.2 shows the business
cycle turning points for both classical and deviation or growth cycles obtained
using this approach. The Centre for Economic Policy Research (CEPR) has
recently formed a committee to set the dates of the euro area business cycle. Its
stated mission is to establish the chronology of recessions and expansions of
the 11 original euro area member countries for 1970–1998 and of the current
euro area as a whole since 1999. The turning points determined by the CEPR
Business Cycle Dating Committee are also indicated in this table. We note that
these are similar to the classical cycles identiﬁed by Artis et al. [21]. Finally,
Table 2.2. Euro Area Business Cycle Turning Points.
Peak Trough
(1) Classical Cycles
∗
1974 (III) 1975 (I)
1980 (I) 1981 (I)
1982 (II) 1982 (IV)
1992 (I) 1993 (I)
(2) Deviation Cycles
∗
1974 (I) 1975 (III)
1976 (IV) 1977 (IV)
1980 (I) 1982 (IV)
1990 (II) 1993 (II)
1995 (I) 1997 (I)
1998 (I)
(3) CEPR Dating Committee
1974 (III) 1975 (I)
1980 (I) 1982 (IIII)
1992 (I) 1993 (III)
∗
Artis, Marcellino, and Proietti [21].
4
See Fagan, Henry, and Mestre [85].
October 9, 2009 15:12 9in x 6in b808-ch02
Facts 23
13.4
13.6
13.8
14.0
14.2
14.4
1970 1975 1980 1985 1990 1995 2000 2005
BPTRENDEURO EUROAREA
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
1970 1975 1980 1985 1990 1995 2000 2005
BPCYCLEEURO
Fig. 2.3. Trend and Cyclical Components in Euro Area GDP.
Fig. 2.3 illustrates the logarithm of real GDP for the euro area and its trend
and cyclical components estimated according to the Baxter–King ﬁlter for the
period 1970(I)–2005(III).
Stock and Watson [199] provide a comprehensive analysis of the volatility
and persistence of business cycles in G7 countries (deﬁned to include the
US, UK, France, Germany, Italy, Japan, and Canada) over the period 1960–
2002. They ﬁnd that the volatility of business cycles has moderated in most
G7 countries over the past 40 years. They also provide evidence on the
synchronization of international business cycles. They base their results on
various measures of correlation of GDP growth across countries. First, they
ﬁnd no evidence for closer international synchronization over their period
of study. This is similar to the ﬁndings of Köse, Prasad, and Terrones [139]
and others. However, in sync with Artis et al. [20], they ﬁnd evidence on
the emergence of two cyclically coherent groups, the eurozone countries and
English-speaking countries (including Canada, the UK, and the US).
Stock and Watson [199] also seek to provide evidence on the sources of
the changes, namely, do they arise from changes in the magnitudes of the
shocks or the nature of the propagation mechanism? Are the sources of the
changes domestic or international? To answer these questions, they use a so-
called factor-structural vector autoregression (FSVAR), which is speciﬁed in
terms of the growth rates of quarterly GDP for the G7 countries. This is a
standard structural vector autoregression (VAR) with an unobserved factor
structure imposed on the VAR innovations. Let Y
t
denote the n ×1 vector of
October 9, 2009 15:12 9in x 6in b808-ch02
24 Business Cycles: Fact, Fallacy and Fantasy
real GDP growth for the G7 countries considered in this study. The standard
VAR model is given by
Y
t
= A(L)Y
t −1
+ν
t
, (3.5)
where the vector of reduced-form errors v
t
has the factor structure
ν
t
= f
t
+
t
, (3.6)
where f
t
is a k ×1 vector with k ≤ n. In this representation, the elements of
t
denote the country-speciﬁc idiosyncratic shocks, and the elements of f
t
denote
the common international shocks. The covariance structure of the shocks is
given by
E(f
t
f

t
) = diag(σ
f 1
, . . . , σ
fk
)
and
E(ν
t
ν

t
) = diag(σ
ν1
, . . . , σ
νk
),
where the elements of f
t
and
t
are assumed to be mutually independent.
Notice that the common shocks affect the output of multiple countries
contemporaneously, whereas the idiosyncratic shocks affect them with a
lag. This provides the identiﬁcation scheme to help identify the common
international shocks. The model allows for spillover effects to occur through
the lagged effect of an idiosyncratic shock. These may arise from the role of
international trade, for example.
Stock and Watson [199] ﬁnd evidence for two common shocks. They
also provide a variance decomposition for the impact of (i) the common
international shocks, (ii) the domestic shocks, and (iii) the spillover effects
of the domestic shocks on the h-step-ahead forecast error for each country.
They consider two sample periods: 1960–1983 and 1984–2002. First, they
ﬁnd that most of the variance of GDP growth can be attributed to common
and idiosyncratic domestic shocks. However, their relative variance varies by
country and by time period. In the ﬁrst period, the impact of international
shocks is estimated to be greatest for countries such as Canada, France, and
Germany, and the least for Italy and Japan. In the second period, domestic
shocks explain almost all of the variance for Japan, reﬂecting the impact of
the ten-year-long deﬂationary episode for this country. Second, the role of
international sources of ﬂuctuations arising from common shocks or from
October 9, 2009 15:12 9in x 6in b808-ch02
Facts 25
spillovers appears to have increased for the US, Canada, and Italy. However,
they also show that the decline in overall volatility of GDP growth for
countries such as the US, Germany, and the UK is due to the decline in
the variance arising from international shocks. Their overall results indicate
that the magnitudes of common international shocks appear to have become
smaller in the 1980s and 1990s than they were in the 1960s and 1970s.
This is the source of the moderation of individual business cycles, and also of
the failure of business cycles to become more synchronized despite the great
increase in trade ﬂows over this period. Finally, they also ﬁnd that shocks have
become more persistent in countries such as Canada, France, and the UK.
2.4. IS THERE A WORLD BUSINESS CYCLE?
Köse, Otrok, and Whiteman [138] consider the issue of a world business
cycle, and use data on 60-odd countries covering seven regions of the world
to determine common factors underlying the cyclical ﬂuctuations in the main
macroeconomic aggregates (output, consumption, and investment) across all
countries and regions as well as across countries and aggregates separately.
They argue that many recent studies of international business cycles focus on
a subset of countries — not the world.
5
Their approach assumes that there are K unobservable factors that
are hypothesized to characterize the dynamic interrelationship among a
cross-country set of economic time series. Let N denote the number of
countries, M the number of time series per country, and T the number of
time periods. Let y
i,t
denote the observable variables for i = 1, . . . , N × M,
t = 1, . . . , T. There are three types of factors:
• N country-speciﬁc factors ( f
country
n
, one per each country);
• R regional factors ( f
region
r
, where r stands for NorthAmerica, LatinAmerica,
Africa, Developed Asia, Developing Asia, Europe, or Oceania);
• the single world factor ( f
world
).
The behavior of each observable variable is related to the factors as follows:
y
i,t
= a
i
+ b
world
i
f
world
t
+ b
region
i
f
region
r,t
+ b
country
i
f
country
n,t
+
i,t
, (4.7)
5
For recent studies, see, for example, Gregory, Head, and Raynauld [109] or Lumsdaine and
Prasad [154].
October 9, 2009 15:12 9in x 6in b808-ch02
26 Business Cycles: Fact, Fallacy and Fantasy
where E(
i,t

j,t
) = 0 for i = j. The coefﬁcients b
j
i
denote the factor loadings,
and show how much of the variation in y
i,t
can be explained by each factor.
Thus, the model assumes that the behavior of M × N time series can be
explained by N + R + 1 factors, where N + R + 1 < MN. Both the
idiosyncratic errors and the factors are allowed to be serially correlated, and to
follow autoregressions of orders p
i
and q
k
, respectively, as

f
k
,t −q
k
+ u
f
k
,t
, (4.10)
where E(u
f
k
,t
u
f
k
,t
) = σ
2
f
k
, E(u
f
k
,t
u
i,t −s
) = 0 for all k, i, s. The innovations
u
i,t
, i = 1, . . . , M × N, and u
f
k
,t
, k = 1, . . . , K , are mean zero,
contemporaneously uncorrelated random variables. Thus, all the covariation
among the observable series y
i,t
is due to the commonfactors, whichthemselves
may be serially correlated.
Since the common factors are unobserved, we cannot identify uniquely
the sign or scale of the factors or the factor loadings. One identiﬁcation device
is to assume that the factor loading on one of the factors is positive. For the
world factor, the sign of the factor loading for US output is considered positive.
Likewise, for the regional factor, the sign of the factor loading corresponding
to North America is considered positive, and the country factors are identiﬁed
by positive factor loadings on the output of each country. Finally, scales
are identiﬁed by assuming that each σ
2
f
k
is equal to a constant. There are
several ways to estimate this model. Since the factors are unobservable, one
approachis to use a Kalmanﬁltering algorithmtogether withclassical statistical
techniques to estimate the model’s parameters; an alternative is to use Bayesian
estimation techniques. This procedure yields a joint posterior distribution
for the unobserved factors and the unknown parameters, conditional on the
data. In what follows, we report results based on the median of the posterior
distribution for the factors, and defer a discussion of the estimation techniques
to Chapter 7.
October 9, 2009 15:12 9in x 6in b808-ch02
Facts 27
Köse et al.’s paper [138] shows that the world factor identiﬁes many of
the major cyclical events over a 30-year period: the expansionary periods of
the 1960s, the recession of the 1970s associated with the ﬁrst oil shock, the
recession of the 1980s associated with the debt crisis in developing countries
and the tight monetary policies in the major developed countries, and the
downturn and recession of the early 1990s. However, in contrast to models
estimated using a smaller sample of countries, the world factor based on a large
set of developed and developing countries implies that the recession of the
1980s was, if anything, as severe as the recession of the mid-1970s. Recall that
the world, regional, and country factors are modeled as autoregressive processes
which may display substantial persistence depending on the estimated values
of the parameters. A second important ﬁnding from the paper is that most of
the persistent co-movement across countries and aggregates is captured by the
world factor. By contrast, the regional and country factors explain covariation
or co-movement that is less persistent.
The paper also allows for a simple decomposition of variance attributed to
the different factors. Using the representation of the model, we have that
Var(y
i,t
) = (b
world
i
)
2
Var(f
world
t
) + (b
region
i
)
2
Var(f
region
r,t
)
+(b
country
i
)
2
Var(f
country
n,t
) + Var(
i,t
). (4.11)
Thus, the fraction of variance attributed to the world factor can be expressed as
(b
world
i
)
2
Var(f
world
t
)
Var(y
i,t
)
.
They ﬁnd evidence that, ﬁrst, the world factor explains nearly 15%of variation
in output growth, 9%of consumption growth, and 7%of investment growth.
They interpret this ﬁnding as evidence for a world business cycle. Second, the
world factor is more successful in explaining economic activity in developed
relative to developing countries. Third, country factors account for a much
larger fraction of consumption growth than output growth. This ﬁnding has
been documented further in the international business cycle literature, which
is discussed in detail in Chapter 4.
The authors also document some noteworthy ﬁndings regarding the
sources of volatility of investment growth. Speciﬁcally, they ﬁnd that country
and idiosyncratic factors account for a much larger share of variability in
investment growth than the world and regional factors. Second, idiosyncratic
October 9, 2009 15:12 9in x 6in b808-ch02
28 Business Cycles: Fact, Fallacy and Fantasy
shocks explain a much larger fraction of the volatility in investment growth in
developing countries relative to developed countries. The authors pose this as
a puzzle. However, much work on investment behavior shows that irreversible
investment decisions are particularly susceptible to risk and uncertainty.
6
Altug, Demers, and Demers [10] argue that an important source of risk
for developing countries is political risk or risk arising from the threat of
expropriation, disruptions to market access, policy reversals, and debt and
currency crises.
7
We conjecture that variability in investment due to such
factors is a key channel for inducing idiosyncratic volatility in developing
economies’ investment behavior.
The last two ﬁndings of the paper imply that regional factors play a minor
role in aggregate output ﬂuctuations. Paralleling this ﬁnding, there is little
evidence that the volatility of European aggregates can be attributed to the
European regional factor. This last result is interpreted as evidence against
the existence of a European business cycle. Yet this ﬁnding could also be due
to model misspeciﬁcation. As Stock and Watson [199] note, the dynamic
factor model suffers from the shortcoming that all covariation is attributed
to the common factors. Yet, there is also the possibility that some observed
covariation could result from the spillover effects of idiosyncratic or country
shocks, especially in an era of increased trade ﬂows and ﬁnancial integration.
2.5. HISTORICAL BUSINESS CYCLES
Basu andTaylor [31] examine business cycles within an international historical
perspective. They consider the time series behavior of output, prices, real wages,
exchange rates, total consumption, investment, and the current account for
15 countries including the US, the UK, and other European countries plus
Argentina for the period since 1870. They divide this period into four periods
that also reﬂect the monetary and capital account regimes prevailing in them.
• 1870–1941: This era corresponded to the classic gold standard. It featured
ﬁxed exchange rates and worldwide capital market integration.
6
For recent reviews, see Caballero [53] or Demers, Demers, and Altug [79].
7
In a related literature, Rodrik [178] emphasizes the importance of political risk (in his case, the risk of
policy reversal) for irreversible investment decisions. Several studies document the importance of political
risk in the unsuccessful recovery of private investment following the adoption of IMF stabilization packages
in various countries. See, for example, Serven and Solimano [189].
October 9, 2009 15:12 9in x 6in b808-ch02
Facts 29
• 1919–1939: During this period, the world economy shifted from a
globalized regime to an autarkic regime. This period also corresponded to
the Great Depression.
• 1945–1971: The third regime is the Bretton Woods era, which
corresponded to the post-World War II era of reconstruction and a
resumption of global trade and capital ﬂows.
• Early 1970s–present: This period features a ﬂoating exchange regime and
a period of increasing globalization.
Basu and Taylor [31] argue that considering such a breakdown allows for
the analysis of the impact of different regimes on cyclical phenomena. It also
provides a way to identify the importance of demand- versus supply-side factors
or shocks and the role of alternative propagation mechanisms such as price
rigidity. For example, most explanations of the Great Depression attribute
the source of this massive downturn, which simultaneously occurred in a
number of countries, to monetary phenomena. Consideration of alternative
historical periods, including the era of the Great Depression, thus allows
for an examination of such mechanisms as nominal rigidities in propagating
shocks. Basu and Taylor [31] also argue that their periodization allows for an
analysis of international capital ﬂows and capital mobility in affecting cyclical
ﬂuctuations.
Bordo and Heibling [44] ask whether national business cycles have become
more synchronized. They examine business cycles in 16 countries during
the period 1880–2001, and consider the four exchange rate regimes also
examined by Basu and Taylor [31]. The countries that they examine are
Australia, Canada, Denmark, Finland, France, Germany, Italy, Japan, the
Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, the UK, and
the US. Synchronization refers to the notion that “the timing and magnitudes
of major changes in economic activity appear increasingly similar.” Among
other concepts, they use a statistical measure of synchronization developed
by Harding and Pagan [115], the so-called concordance correlation, which
examines whether the turning points in the different series occur at similar
dates. Instead of using information on NBER-type reference dates, Bordo
and Heibling [44] use real GDP or industrial production series to determine
synchronization. They also use standard output correlations and factor-based
measures.
October 9, 2009 15:12 9in x 6in b808-ch02
30 Business Cycles: Fact, Fallacy and Fantasy
To brieﬂy describe the concordance correlations, let S
it
and S
jt
denote
binary-cycle indicator variables which assume a value of 1 if the economy
is in an expansion, 0 otherwise, for countries i and j. A simple measure of
concordance is deﬁned by the variable
I
ij
=
1
T
T

t =1
[S
it
S
jt
− (1 − S
it
)(1 − S
jt
)]. (5.12)
Let
¯
S
i
= (1/T)

T
t =1
S
it
denote the estimated probability of being in an
expansion, say; then under the assumption that S
it
and S
jt
are independent,
the expected value of the concordance index is 2
¯
S
i
¯
S
j
+1−
¯
S
i
−
¯
S
j
. Subtracting
this from I
ij
yields the mean-corrected concordance index:
I
∗
ij
= 2
1
T
T

t =1
(S
it
−
¯
S
i
)(S
jt
−
¯
S
j
). (5.13)
To derive the concordance correlation coefﬁcient, we divide I
∗
ij
by a consistent
estimate of its standard error under the null hypothesis of independence. Note
that the variance of I
∗
ij
is given by
Var(I
∗
ij
) =
4
T
2
E

T

t =1
(S
it
−
¯
S
i
)(S
jt
−
¯
S
j
)

.
If the two cycles are perfectly synchronized (so that S
it
= S
jt
for all t ), then
the standardized index equals 1; if they are in different states in each period
(so that S
it
= 1 −S
jt
), then the standardized index equals −1; and if the two
cycles are unrelated, the standardized index equals 0.
Bordo and Heibling [44] calculate the business cycle indicators S
it
for
each of the four exchange rate regimes. They deﬁne a recession as one or
more consecutive years of real GDP growth, while an expansion is deﬁned
as one or more years of positive GDP growth. They argue that S
it
= 1
for most countries during the Bretton Woods era of 1948–1972 and hence
leave this period out. For the gold standard era, they ﬁnd that the average
of the correlation coefﬁcients is zero, as half of all pairs of business cycles
are negatively related to each other while the other half are positively related.
In the interwar and post-Bretton Woods era, more than half of all national
business cycles become positively related to each other. The key ﬁnding is
that during the classical gold standard, cycles were, on average, uncorrelated
October 9, 2009 15:12 9in x 6in b808-ch02
Facts 31
with each other; whereas beginning with the interwar period, they started
becoming synchronized with each other. The authors also examine standard
output correlations, which measure the magnitude as well as the direction of
output changes. They ﬁnd that there has been a tendency for higher, more
positive bilateral output correlations by era. They also ﬁnd higher output
correlations for the core European countries (the EEC) and the Continental
Europeancountries. Finally, as inStockandWatson[199], they ﬁndanincrease
in correlations for the Anglo-Saxon countries.
The authors also estimate a so-called static approximate factor model for
the growth rates of GDP. In this model, there are no dynamics in the relation
between output growth rates and the factors, and the idiosyncratic shocks
are allowed to be serially correlated and heteroscedastic. They ﬁnd that the
variability of output due to variability in the common factors has “doubled
from about 20 percent during the Gold Standard era to about 40 percent
during the modern era of ﬂexible exchange rates.” They also estimate restricted
versions of a FSVAR model along the lines of Stock and Watson [199] to
determine the role of common shocks, idiosyncratic shocks, and spillover
effects in generating this result. They consider a so-called center country version
of this model, and a trade linkages version. Inthe former, lagged GDPgrowth of
the center country is included alongside the lagged value of the own country’s
GDP growth in the VAR representation. In the trade linkages version, lagged
GDP growth of the major trading partner is included in place of the center
country’s. They ﬁnd that both global (common) shocks and transmission
have become more important. However, the importance of transmission for
peripheral countries arises only in the trade model, suggesting that it is not
transmission from the center country that accounts for the increased role of
transmission.
Bordo and Heibling [44] conclude by noting that global shocks are the
dominant inﬂuence across all regimes, and that the increasing importance of
global shocks reﬂects the forces of globalization, especially the integration of
goods and services through international trade and the integration of ﬁnancial
markets. Eichengreen and Bordo [84] examine the nature of crises in two
periods of globalization, before 1914 and after 1971. They argue that banking
crises were less severe in the period before 1915, but this was not so for ﬁnancial
or twin crises. Typically, such crises have ﬁgured importantly in emerging
market output ﬂuctuations. We discuss emerging market business cycles in
detail in Chapter 6.
October 9, 2009 15:12 9in x 6in b808-ch02
This page intentionally left blank This page intentionally left blank
October 9, 2009 15:12 9in x 6in b808-ch03
Chapter 3
Models of Business Cycles
The standard approach to classifying business cycle models follows Ragnar
Frisch’s [95] terminology of impulses and propagation mechanisms. The
shocks or impulses thought to instigate business cycles have typically been
varied and diverse. Technology shocks which alter a society’s production
possibilities frontier, whether they are permanent or transitory such as oil
shocks, have ﬁgured prominently in the recent literature. Weather shocks
have always had a place among the impulses thought to trigger ﬂuctuations
in economic activity. Political shocks, wars, and other disruptions in market
activity have also been acknowledged to play a role. More controversially, one
could also assign a role to taste shocks or changes in preferences of consumers.
Equivalently, one could argue, as Keynes [127] did, that investors’ “animal
spirits” or, more precisely, changes in their subjective beliefs could help to
trigger business cycles.
Much of the controversy in the business cycle literature has stemmed from
differences attached to the importance of alternative propagation mechanisms
and the associated shocks. Current real business cycle (RBC) theory argues that
business cycles canarise infrictionless, perfectly competitive, complete markets
in which there are real or technology shocks. This approach emphasizes the
role of intertemporal substitution motives in propagating shocks. By contrast,
Keynesian and New Keynesian models stress the role of frictions such as price
stickiness. In a simple labor market model, if real wages do not adjust downward
when there is a negative shock to demand, the result is unemployment and
greater declines in output relative to a situation with ﬂexible prices. The Great
33
October 9, 2009 15:12 9in x 6in b808-ch03
34 Business Cycles: Fact, Fallacy and Fantasy
Depression and the recent ﬁnancial crises also indicate the importance of credit
market frictions and ﬁnancial accelerator-type effects as a potential propagation
mechanism for business cycles. In this case, shocks originating in various
ﬁnancial markets lead to a widening circle of bankruptcies and bank failures
as well as large and negative effects on real output.
1
In this chapter, we discuss variants of the RBC model. Since the debate
has revolved around the efﬁcacy of technology shocks and intertemporal
substitution effects in replicating business cycles, it seems imperative to discuss
the mainpropagationmechanisms of the RBCmodel. Inthe following chapter,
we discuss an international RBC model to describe how the transmission
mechanisms proposed by this approach translate into an open-economy
setting.
2
3.1. AN RBC MODEL
RBC theory has become popular in recent years because its micro foundations
are fully speciﬁed and it links the short run with the neoclassical growth model.
The prototypical RBC model has the structure of a standard neoclassical
growth model with a labor/leisure choice incorporated. As Kydland and
Prescott argue, this is a crucial modiﬁcation as “more than half of business cycle
ﬂuctuations are accounted for by variations in the labor input” ([143], p. 173).
There is a representative consumer who derives utility from consumption and
leisure, and a constant-returns-to-scale (CRTS) production technology for
producing the single good using labor and capital. All markets are perfectly
competitive, all factors are fully employed, and all prices adjust instantaneously
to clear markets. A number of these assumptions have come under criticism
in the recent literature. The standard RBC model also assumes that there is no
government, that households consume only goods produced in the market,
and that there is no trade with the rest of the world. Some of the extensions
of the RBC approach have modiﬁed the basic framework to incorporate the
role of government, home production, and international trade. We will discuss
these extensions in the following sections.
1
See, for example, Bernanke and Gertler [40] or Kiyotaki and Moore [134].
2
Rebelo [177] provides a recent review of alternative models and directions associated with the RBC
agenda, but his discussion is abbreviated to satisfy the constraints imposed by a journal article.
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 35
Turning to the basic model, the representative agent has time-separable
preferences over consumption and leisure choices given by
U = E
0
_
∞

t
) = . This is nowan optimal control problem
with a quadratic objective and linear constraints. Such problems can be solved
using the methods for solving dynamic optimization problems that satisfy a
certainty equivalence property; in other words, the solution for the stochastic
optimal control problem is identical to the solution for the deterministic
version of the problem with the shocks ε
t +1
replaced by their expectation
E(ε
t +1
). This class of problems is known as optimal linear regulator problems.
Ljungqvist and Sargent [146] and Anderson, Hansen, McGrattan, and Sargent
[16] provide further discussion of the formulation and estimation of linear
dynamic economic models.
Bellman’s equation for this problem is given by
V (x
t
) = max
u
t
_
x

). (2.29)
Notice that the shocks ε
t +1
do not affect the optimal choice of u
t
. Substituting
for u back into the deﬁnition of V (x) yields
P = R + βA

PA − (βA

PB + W )(Q + βB

PB)
−1
(βB

PA + W

),
d = (1 − β)
−1
[βE(ε

Pε)].
The equation for P is known as the algebraic matrix Riccati equation for
so-called optimal linear regulator problems.
6
The solution for P can be
obtained by iterating on the matrix Riccati difference equation:
P
n+1
= R + βA

).
In the RBC literature, the model is then calibrated with the data.
Calibration refers to the practice of determining the parameters of the model
based on its steady state properties and the results of other studies, given
particular functional forms for the productionfunctionandthe utility function
(see Cooley and Prescott [75]). A stochastic process for z
t
is also speciﬁed and
a random number generator is used to simulate the TFP time series. The
simulated series are used to generate time series for consumption, output,
5
To derive this result, we have used the rules for differentiating quadratic forms as
∂u

Qu
∂u
= [Q + Q

]u = 2Qu,
∂x

Tu
∂u
= T

x and
∂u

Tx
∂u
= Tx.
6
Notice that the matrices R, Q, A, and B must be further restricted to ensure the existence of a solution
to the matrix difference equation deﬁning P
n
. One condition that sufﬁces is that the eigenvalues of A are
bounded in modulus below unity.
October 9, 2009 15:12 9in x 6in b808-ch03
44 Business Cycles: Fact, Fallacy and Fantasy
investment, and labor. Trends in the data are removed using a given ﬁlter. The
means, variances, covariances, and cross-correlations for the simulated time
series are compared to the comparable statistics in the data. The model is then
judged to be close or not close based on this comparison, so the approach
differs signiﬁcantly from standard econometrics.
We now illustrate this solution method for the standard RBC model with
a labor-leisure choice presented at the beginning of this section. Consider the
parameter values β = 0.95, α = 1/3, θ = 0.36, δ = 0.10, ρ = 0.95, and
σ = 0.028. The steady values for the variables are given by
¯
k = 1.1281,
¯
i = 0.1128, ¯ y = 0.4783, ¯ c = 0.3655, and
¯
h = 0.2952. We ﬁrst calculate a
set of unconditional moments that have been used in the RBC literature to
match the model with the data. These are obtained by simulating the behavior
of the different series across a given history of the shock sequence. Speciﬁcally,
we draw a sequence of shocks {ˆ
t +1
}
2999
t =0
that are normally distributed
with mean zero and standard deviation 0.028, and generate a sequence of
technology shocks beginning from some initial value z
0
as ˆ z
t +1
= ρz
t
+ˆ
t +1
.
We then use the linear decision rules for all variables to simulate for the
endogenous variables based on the same history of shocks. Finally, we calculate
unconditional moments for the different series after dropping the ﬁrst 1000
observations.
The typical set of unconditional moments used in the RBC literature is
displayed in Table 3.1. We ﬁnd that consumption ﬂuctuates slightly less than
output and investment ﬂuctuates signiﬁcantly more. Likewise, productivity is
almost as variable as output. By contrast, we ﬁnd that the variation in hours is
typically quite low. In terms of the correlation of each series with output, we
ﬁnd that all series are procyclical, with hours showing the least procyclicality.
The RBC theorists have interpreted these results as implying that, in the
ﬁrst instance, the model is capable of delivering some of the salient features
Table 3.1. Cyclical Properties of Key Variables.
Standard Deviation (%) Correlation with Output
Output 8.8616 1
Consumption 8.1134 0.9163
Investment 12.9238 0.9807
Hours 1.6684 0.4934
Productivity 7.7496 0.9840
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 45
of the data. Speciﬁcally, the model predicts that consumption ﬂuctuates less
than output whereas investment ﬂuctuates more, as we observe in the data.
Consumption, investment, and productivity are all strongly procyclical. In
terms of the variation accounted by the technology shocks, the technology
shock explains much but not all of the variation in output. Thus, the RBC
model has been viewed as being able to generate cycles endogenously and
having economic signiﬁcance even if the calibration method is controversial.
This is discussed further in the following sections.
Figure 3.2 illustrates the response of all the variables to a 1% shock in
technology starting fromthe steady-state capital stock. We note that hours and
output both increase and then fall back to a lower level. The percentage change
in output exceeds the percentage change in the technology shock because
optimal hours also show a positive response to the technology shock. By
contrast, capital and consumption showa humped-shaped response, rising ﬁrst
and thendeclining back to a lower level. Consumptionshows a gradual positive
response because investment is initially high. These responses have been widely
0 5 10 15 20
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Years after shock
%

d
e
v
i
a
t
i
o
n
s

f
o
r
m

s
t
e
a
d
y

s
t
a
t
e
Hours
Output
Consumption
Capital
Technology
Fig. 3.2. Impulse Responses to a Shock in Technology.
October 9, 2009 15:12 9in x 6in b808-ch03
46 Business Cycles: Fact, Fallacy and Fantasy
documented inthe RBCliterature (see, for example, Uhlig [206]). The positive
response of hours toa positive productivity shockhas alsobecome animportant
point of difference between models that assume perfectly ﬂexible prices versus
those that assume nominal price rigidity. We discuss New Keynesian models
with monopolistic competition and prices that are ﬁxed in the short run in
Chapter 5.
3.3. INITIAL CRITICISMS
In perhaps what is the most famous example of an RBC model, Kydland and
Prescott [141] formulated a real business cycle model in which preferences
are not separable over time with respect to leisure and there exists a time-to-
build feature in investment for new capital goods. The production function
displays constant returns to scale with respect to hours worked and a composite
capital good. The only exogenous shock to their model is a randomtechnology
shock whichfollows a stationary ﬁrst-order autoregressive process. They use the
quadratic approximation procedure to obtain linear decision rules for a set of
aggregate variables, and generate time series for the remaining series by drawing
realizations of the innovation to the technology shock. They calculate a small
set of moments associated with each series to match the model with the data.
When calibrating their model, Kydland and Prescott choose the variance
of the innovation to the technology shock to make the variability of the output
series generated by their model equal to the variability of observed GNP. As
McCallum [156] notes, this feature of their analysis makes it difﬁcult to judge
whether “technology shocks are adequate to generate output, employment, etc.
ﬂuctuations of the magnitude actually observed.” One of the most pervasive
criticisms of the model is that there is no evidence of large, economy-wide
disturbances that can play the role of the technology shocks posited by
Kydland and Prescott. The only exception is oil price shocks, leading one
to question whether the model was developed in reaction to the supply-side
disturbances that characterized the 1970s. The RBC approach in general has
difﬁculty in answering what some other examples of big shocks are. Summers
[202] argues that “the vast majority of what Prescott labels technology shocks
are in fact observable concomitants of labor hoarding and other behavior
which Prescott does not allow for in his model.” This point was subsequently
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 47
elaborated on by Hall [110, 111], who argued that the procyclical movements
in the Solow residual were, in fact, endogenous changes in efﬁciency arising
from labor hoarding, increasing returns to scale, or price markups.
Summers [202] also took issue with the parameters of the model that
Kydland and Prescott [141] or Prescott [173] advocated in their calibration
exercise. He argued that the econometric evidence presented by Eichenbaum,
Hansen, and Singleton [83] on the intertemporal substitution of leisure
hypothesis pointed to a share of time allocated to market activities that was
more in the range of one-sixth, not one-third. He also noted that a real interest
rate of 4% was inconsistent with the historical evidence of very low yields
averaging around 1%. Singleton [192] raised the problem of inference and
sampling uncertainty surrounding the various measures of ﬁt proposed in
the RBC literature. He noted that the calibration approach did not provide
a straightforward and transparent way of judging whether a given model
constituted an improvement over another model. Eichenbaum [82] raised
the issue of the model’s ﬁt based on the ratio of variance for GDP implied
by the model and the data. Eichenbaum showed that the standard error of
this ratio was heavily inﬂuenced by the parameters of the assumed stochastic
process for the technology shock.
The policy implications of the RBC model have also come under attack.
According to the RBC model, business cycle ﬂuctuations are Pareto optimal
and there is no role for the government to try to smooth or mitigate
the ﬂuctuations. In fact, such policy efforts are inefﬁcient. Prescott [173]
commented as follows:
Economic theory predicts that, given the nature of the shocks to technology
and people’s willingness and ability to intertemporally and intratemporally
substitute, the economy will display ﬂuctuations like those the U.S. economy
displays. . . . Indeed, if the economy did not display the business cycle
phenomena, there would be a puzzle.
Thus, according to this approach, cyclical ﬂuctuations are viewed as the natural
response of the economy to changes that affect individuals’ consumption or
productiondecisions. However, if prices or wages are ﬁxedor rigidor if there are
other types of frictions arising fromcredit markets, thenthe policy implications
of the RBC approach may not be valid and the economy may operate with
high levels of unemployment and excess capacity over long periods. In this
October 9, 2009 15:12 9in x 6in b808-ch03
48 Business Cycles: Fact, Fallacy and Fantasy
case, government policies may be required to move the economy towards a
full employment level.
In the chapters that follow, we will discuss the efﬁcacy of the RBC model
in serving as a model of aggregate ﬂuctuations as well as questions that have
lingered to this day about the adequacy of the calibrationapproachinmatching
the model to the data.
3.4. “PUZZLES”
Many of the initial criticisms which had been voiced by opponents of the RBC
approach took on the character of “puzzles”. These puzzles have constituted
the basis of much further research in the area.
We can list some of these puzzles as follows:
• The variability of hours and productivity: As seen in Table 3.1, one of the
main problems with the standard RBC model is that it cannot capture the
variation in the aggregate labor input (see also Kydland [140]). In the data,
employment is strongly procyclical and almost as variable as output while
real wages are weakly procyclical. In the standard model, a productivity
shock shifts the marginal product of labor so that the observed variations in
employment can only occur if the labor supply curve is relatively elastic. Yet,
micro studies ﬁnd that wage elasticity of labor supply is quite low. In this
case, the marginal productivity shock should lead to most of the adjustment
in real wages and less in the quantity of labor. Furthermore, in the data,
hours of work per worker adjusts very little over the cycle. About two-thirds
of the variability in total hours worked comes from movements into and
out of the labor force, and the rest is due to adjustment in the number of
hours worked per employee. These ﬁndings create a puzzle for the standard
RBC model.
• The productivity puzzle: In the data, the correlation between productivity
and hours worked is near zero or negative, while the correlation between
productivity and output is positive and around 0.5.
7
By contrast, the
RBC model, which is driven entirely by productivity shocks, generates
correlations that are large and positive in both cases. Another problem that
7
See, for example, Christiano and Eichenbaum [65], who measure hours worked and productivity
based on both household and establishment-level surveys conducted by the US Department of Labor.
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 49
arises in matching the model and the data is that in the data, labor’s share
of income moves countercyclically, whereas in the RBC model labor’s share
is ﬁxed.
• Reverse causality: The stylized facts of business cycles state that money,
especially M2, appears to be a leading indicator of aggregate output.
However, this is inconsistent with the notion that TFPshocks are the driving
force behind business cycle activity. The conventional wisdomregarding the
money-output correlation, summarized by Friedman and Schwartz [94] in
their study of monetary history, is that the causality goes from money to
output but with “long and variable lags”.
3.4.1. A Model with Indivisible Labor Supply
The indivisible labor, lottery models studied by Gary Hansen [112] and
Richard Rogerson [179] have been used to explain the stylized fact that
aggregate hours vary more than productivity does. This framework also
provides a way for reconciling large labor supply elasticities at the aggregate
level with low labor supply elasticities at the individual level. This is
accomplished by assuming that individuals can work all the time or not at all.
To account for the nonconvexities introduced by the work/nonwork decision,
it is assumed that individuals choose the probability of working, π
t
. A lottery
then determines whether an individual actually works. This economy is one in
which individuals and a ﬁrm trade a contract that commits the household to
work h
0
hours with probability π
t
. Since what is being traded is the contract,
the individual gets paid regardless of whether he works or not. We nowdescribe
a version of the RBC model due to Rogerson [179] and Hansen [112] that
allows for ﬁxed costs and nonconvexities in labor supply.
Suppose that workers are constrained to work either zero or
ˆ
h hours, where
0 <
ˆ
h < 1. (4.30)
The main idea is that there are nonconvexities or ﬁxed costs that make varying
the number of employed workers more efﬁcient than varying hours per worker.
Let π
t
denote the probability that a given agent is employed in period t so
that the number of per capita hours worked is given by
H
t
= π
t
ˆ
h. (4.31)
October 9, 2009 15:12 9in x 6in b808-ch03
50 Business Cycles: Fact, Fallacy and Fantasy
Let c
0,t
denote the consumption of an unemployed worker and c
1,t
denote
the consumption of an employed agent. Then the expected utility of the
representative consumer, taking into account the work versus nonwork
decision, is given by
E[u(c
t
, l
t
)] = π
t
u(c
1,t
, 1 −
ˆ
h) + (1 − π
t
)u(c
0,t
, 1).
Assume that the individual utility function has the form
u(c, l ) = ln (c) + A ln (l ). (4.32)
The social planner solves the problem
max
π
t
,c
0,t
,c
1,t
E[u(c
t
, l
t
)] s.t. π
t
c
1,t
+ (1 − π
t
)c
0,t
= c
t
.
Notice that the social planner chooses the consumption allocations of each
agent plus the probability of their working. When agents do work, they
must supply
ˆ
h hours of work so that there is no choice over hours of work
directly. Let λ
t
denote the Lagrange multiplier on the feasibility constraint
for consumption. Omitting the work/leisure decision for the moment, the
ﬁrst-order conditions with respect to (c
0,t
, c
1,t
) are
1 − π
t
c
0,t
= (1 − π
t
)λ
t
, (4.33)
π
t
c
1,t
= π
t
λ
t
. (4.34)
It follows that c
0,t
= c
1,t
= c
t
so that the agent consumes the same amount
whether or not he is working. Hence, the unemployed worker enjoys higher
utility since working causes disutility. In this model, ex ante all individuals are
alike, but ex post they differ because some work while others enjoy leisure. With
complete insurance and identical preferences that are separable with respect
to consumption and leisure, all individuals have the same consumption but
the unemployed are better off. This is a feature that is counterfactual to the
working of actual labor markets.
Notice that the agent will consume c
t
whether or not he is working. Hence,
expected utility (where the expectation is over whether or not you work) is
ln (c
t
) + π
t
A ln (1 −
ˆ
h) + (1 − π
t
)A ln (1), (4.35)
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 51
where A is a positive constant. Using the deﬁnition of H
t
, π
t
= H
t
/
ˆ
h. Now
substitute for π
t
in (4.35) and use ln (1) = 0 to obtain
E[u(c
t
, l
t
)] = ln (c
t
) + π
t
A ln (1 −
ˆ
h)
= ln (c
t
) − BH
t
, (4.36)
where
B =
−A ln (1 −
ˆ
h)
ˆ
h
.
Comparing equation (4.32) with equation (4.36) shows the effect of the
lottery assumption. The former speciﬁcation for preferences implies a low
intertemporal elasticity of substitution in labor supply, which is consistent with
assumptions about individual behavior. By contrast, the latter speciﬁcation —
which is linear in total hours H
t
— implies a high intertemporal elasticity at
the aggregate level.
The preferences can now be written as
U = E
0
_
∞

t =0
β
t
u(c
mt
, c
ht
, h
mt
, h
ht
),
where 0 < β < 1. In this expression, c
mt
is the consumption of a market
good; c
ht
is consumption of the home-produced good; h
mt
is labor time spent
in market work; and h
ht
is labor time spent in home work. Assume that
u
1
> 0, u
2
> 0, u
3
< 0, and u
4
< 0. The total amount of time available to
the household is normalized as unity, and leisure is deﬁned as time not spent
working in the market or at home:
l
t
= 1 − h
mt
− h
ht
.
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 55
At each date, the household can purchase market goods c
mt
, market capital
goods k
mt
, and household capital goods k
ht
. Household capital goods are
used in home production, but market capital goods are rented to ﬁrms at the
competitive rental rate r
t
. Letting w
t
denote the wage rate and δ
m
and δ
h
denote the depreciation rates on market and home capital, respectively, the
household’s budget constraint is
c
mt
+ k
m,t +1
+ k
h,t +1
≤ w
t
h
mt
+ r
t
k
mt
+ (1 − δ
m
)k
mt
+ (1 − δ
h
)k
ht
.
Home goods are produced according to the home production function:
c
ht
= g(h
ht
, k
ht
, z
ht
),
where z
ht
is a shock to home production and g is increasing and concave in
labor and capital.
Alternative measures put home production to be in the range of 20–50%
of GDP. In the model, home production is used to produce a nontradeable
consumption good. A rise in market productivity may induce households
to substitute away from home production towards market production. This
gives us another margin to substitute market labor and improves the model’s
predictions. Unlike the standard model, the labor supply curve also shifts in
response to a good productivity shock, thereby leading to greater variability
in labor. However, one criticism of the home production theory is that it
suggests that all movements out of the labor force (toward home production)
are voluntary.
Labor Hoarding
A third way to resolve the productivity puzzle is through a labor hoarding
argument. This says that the effective labor input can be altered even though
the total number of workers is ﬁxed. The ﬁrm may not alter its work force
every time there is a productivity shock (which would occur through a shift
in the marginal product of labor curve). However, if there are costs to hiring
or laying workers off, then ﬁrms may retain workers even though they are
not exerting much effort. Hence, labor effort is likely to be adjusted ﬁrst in
response to a productivity shock. Eventually more workers may be hired or
ﬁred, but only after longer periods of time.
October 9, 2009 15:12 9in x 6in b808-ch03
56 Business Cycles: Fact, Fallacy and Fantasy
To illustrate the role of labor hoarding, consider a production function for
aggregate output Y
t
which depends on an exogenous technology shock A
t
,
capital K
t
, and a labor input that reﬂects variations in work effort following
Burnside, Eichenbaum, and Rebelo [52]. Speciﬁcally, let f denote a ﬁxed shift
length; N
t
, the total number of workers; and W
t
, the work effort of each
individual. Thus,
Y
t
= A
t
K
α
t
[fN
t
W
t
]
1−α
, 0 < α < 1.
In the standard model, the production function can be written as
Y
t
= S
t
K
α
t
[H
t
]
1−α
,
where H
t
denotes the total hours worked. In the absence of variations in work
effort, H
t
= fN
t
. Notice that the conventionally measured Solow residual S
t
is related to true technology shock A
t
as
ln (S
t
) = ln (A
t
) + αln (K
t
) + (1 − α)[ln (f ) + ln (N
t
) + ln (W
t
)]
−αln (K
t
) − (1 − α)[ln (f ) + ln (N
t
)]
= ln (A
t
) + (1 − α) ln (W
t
).
Decisions to alter effort levels will be the outcome of a maximizing decision, so
that the movements in the Solow residual are not entirely exogenous. If labor
hoarding is included in the model, then the productivity/hours correlation is
reduced and more closely matches the data. The comments made regarding
ﬂuctuations in the effort level of labor also apply to capital. The measured
capital in the Solow residual does not take into account optimal ﬂuctuations
in capital utilization rates. This can lead to variation in the Solow residual that
is not related to changes in productivity or technology.
3.4.3. Reverse Causality
One approach to dealing with this criticism is to introduce money and
banking into the standard RBC model following King and Plosser [129].
These authors observe that transactions services (as provided by money and
the banking sector) can be viewed as an intermediate input that reduces the
cost of producing output and, hence, can be treated as a direct input into the
aggregate production function. To brieﬂy describe their framework, suppose
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 57
the ﬁnal good is produced according to the production technology
y
t
= f (k
ft
, n
ft
, d
ft
)φ
t
, (4.49)
where k
ft
is the amount of capital, n
ft
is the amount of labor services, and
d
ft
is the amount of transactions services used in the ﬁnal goods industry. In
this expression, φ
t
is a shock to production of the ﬁnal good at time t . The
ﬁnancial industry is assumed to provide accounting services that facilitate the
exchange of goods by reducing the amount of time that would be devoted to
market transactions. The production of the intermediate good is given by
d
t
= g(n
dt
, k
dt
)λ
t
, (4.50)
where n
dt
and k
dt
denote the amounts of labor and capital allocated to the
ﬁnancial sector, respectively, and λ
t
captures technological innovations to
the ﬁnancial services industry. Households maximize the expected discounted
value of utility from consumption c
t
and leisure l
t
as
E
0
_
∞

< 0. The household chooses an amount of transactions
services d
ht
so as to minimize the total transactions costs, w
t
n
τt
+ ρ
t
d
ht
,
where w
t
is the real wage and ρ
t
is the rental price of transactions services.
This implies a demand for transactions services that can be obtained from the
ﬁrst-order condition
ρ
t
= w
t
τ

)
−1
. Likewise, hours allocated to
producing transactions services is n
∗
τt
= τ
∗
(h(ρ
t
/w
t
))(c
t
+ i
t
). Finally, we
October 9, 2009 15:12 9in x 6in b808-ch03
58 Business Cycles: Fact, Fallacy and Fantasy
require that the household’s time allocated to the different activities sums to
1, n
t
= n
ft
+ n
dt
+ n
τt
.
This framework can be used to rationalize the observations regarding
money and output. Speciﬁcally, inside money, or a broad measure of money
that includes commercial credit, is more closely related to output than outside
money. Suppose that a shock to the production of ﬁnal goods or, equivalently,
a positive shock to productivity occurs. Then, consumption and leisure of the
representative consumer will rise but so will investment demand as consumers
seek to spread the extra wealth over time. If the substitution effect of an increase
in the marginal product of labor outweighs the wealth effect of the productivity
shock, there will be an increase in hours of work. As a consequence, investment
demand rises and output will also increase in response to the additional hours
worked, so ﬁrms will wish to ﬁnance a greater volume of goods in process.
As a result, commercial credit, or inside money, responds to the positive
productivity shock. The increase in output will also stimulate the demand
for transactions services by households and ﬁrms. Hence, the causality runs
from the productivity shock to money even though the increase in money
occurs before the higher productivity shock.
Ahmed and Murthy [3] provide a test of this hypothesis using a small open
economy such as Canada to evaluate the impact of exogenously given terms
of trade and real interest rates versus domestic aggregate demand and supply
disturbances. Their results are derived from a structural vector autoregression
(VAR) with long-run restrictions to identify the alternative structural shocks.
Consistent with the RBC view, they ﬁnd that an important source of the
money-output correlation is output shocks affecting inside money in the
short run. Another possibility is that the central bank uses the accommodative
monetary policy.
9
During an expansion, if interest rates are rising and ﬁrms wish
to invest more in anticipation of higher expected proﬁts, the central bank may
expand the money supply to keep interest rates from rising too rapidly.
3.5. THE SOURCE OF THE SHOCKS
The RBCmodel has come under much criticismfor assuming that technology
or productivity shocks are the sole driving force of cyclical ﬂuctuations. One
9
See Sims [190].
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 59
way of dealing with this criticism is to introduce alternative sources of shocks
into RBCmodels. In the previous section, we discussed the role of government
or ﬁscal shocks in generating observed co-movements of the data. In this
vein, Christiano and Eichenbaum [65] introduce government consumption
shocks as a way of breaking the strong and positive relationship between hours
worked and productivity implied by the basic RBC model. Other papers that
have employed the RBC framework with ﬁscal shocks include Braun [46] and
McGrattan [158]. These authors consider the impact of distortionary taxes
on real outcomes. War shocks can also be modeled as a source of cyclical
ﬂuctuations (see Barro [27] and Ohanian [169]).
10
3.5.1. Investment-Speciﬁc Technological Shocks
An important source of shocks is the investment-speciﬁc technological change.
This has been examined by Greenwood, Hercowitz, and Krusell [106, 107].
Following Gordon [104], these authors motivate their analysis by noting that,
during the period 1950–1990, the relative price of new equipment declined
by 3% at an average annual rate while the equipment-GNP ratio increased
substantially. Furthermore, there was a negative correlation (−0.46) between a
de-trendedmeasure of the relative price of capital andinvestment expenditures.
The notion behind this type of change is that production of capital goods
becomes increasingly efﬁcient over time, thereby stimulating output growth.
11
In Greenwood et al. [107], the role of investment-speciﬁc technological
shocks is used to account for cyclical ﬂuctuations. Consider a standard RBC
model with a representative consumer who derives utility from consumption
and leisure as
E
0
_
∞

t =0
β
t
[θ ln (c
t
) + (1 − θ) ln (1 − l
t
)]
_
, 0 < θ < 1,
10
Altug, Demers, and Demers [10] examine the impact of political risk and regime changes due to
changes in the party in power on real investment decisions under irreversibility. Although their analysis is
not general equilibrium, they nevertheless show that political events can have non-negligible effects on real
economic variables.
11
In Greenwood, Hercowitz, and Krusell [106], the authors use a growth accounting approach to
show that investment-speciﬁc shocks can account for 60% of postwar US growth of output per man-hour.
October 9, 2009 15:12 9in x 6in b808-ch03
60 Business Cycles: Fact, Fallacy and Fantasy
where c
t
denotes consumption and l
t
denotes labor supply. There is a
production technology that depends on labor and the services from two types
of capital goods, equipment denoted k
e
and structures denoted k
s
. Equipment
is utilized at the time-varying rate h; hence, the service ﬂow is denoted hk
e
.
The production function is given by
y = F(hk
e
, k
s
, l , z) = z(hk
e
)
α
e
k
α
s
s
l
1−α
e
−α
s
, 0 < α
e
, α
s
, α
e
+ α
s
< 1,
where z is a measure of total factor productivity. Structures evolve according
to the standard law of motion:
k

s
= (1 − δ
s
)k
s
+ i
i
, 0 < δ
s
< 1. (5.54)
The law of motion for equipment is given by
k

e
= (1 − δ
e
(h))k
e
+ i
e
q, (5.55)
where the variable q shows the investment-speciﬁc technological change in
equipment and
δ
e
(h) =
b
ω
h, ω > 1. (5.56)
Thus, investment in equipment is subject to variable rates of utilization and
depreciation which are convex in the utilization rate. The shocks to technology
and to the efﬁciency of equipment capital constitute the drivers of the
model. They evolve as z
t +1
= γ
t +1
z
exp (ζ
t +1
) and q
t +1
= γ
t +1
q
exp (η
t +1
),
where ζ
t +1
and η
t +1
are innovations to productivity and investment-
speciﬁc technological change, respectively, that follow ﬁrst-order Markov
processes, and γ
z
and γ
q
denote the average gross growth rates of the two
processes.
Investment in both structures and equipment is also subject to convex costs
of adjustment denoted as a = a
e
+ a
s
, where
a
e
= exp (η)φ
e
(k

e
/q − κ
e
k
e
/q)
2
/(k
e
q), φ
e
, κ
e
> 0, (5.57)
a
s
= φ
s
(k

s
− κ
s
k
s
)
2
, φ
s
, κ
s
> 0. (5.58)
Finally, there is a government which levies taxes on labor and capital income
at the rates τ
l
and τ
k
, respectively. The revenue raised by the government is
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 61
returned to the private sector through lump-sum transfers τ. This yields the
government budget constraint:
τ = τ
k
(r
e
hk
e
+ r
s
k
s
) + τ
l
wl , (5.59)
where r
e
and r
s
denote the rental rate onequipment and structures, respectively,
and w denotes the real wage rate. The resource constraint requires that
consumption plus investment in the two types of capital equal output net
of adjustment costs:
c + i
e
+ i
s
= y − a
e
− a
s
. (5.60)
In this analysis, the investment-speciﬁc technological shock plays a key role.
A higher value of q directly affects the quantity of equipment that will be
available to produce output next period. However, it also affects the utilization
cost of existing equipment this period by lowering the replacement cost of old
capital goods, and hence increases its utilization. Hence, investment-speciﬁc
technological change increases the services from old equipment at the same
time as it increases the quantity of equipment available for next period.
We note that the economy displays balanced growth as z
t
and q
t
grow at
the rates γ
z
and γ
q
, respectively. In the balanced growth path equilibrium,
the quantity of labor l and the utilization rate h are constant. The resource
constraint and the accumulation equation for structures imply that y, c, i
e
,
i
s
, a
e
, a
s
, and k
s
all have to grow at the same rate; denote this rate by g.
Equipment, however, grows faster, at the rate g
e
= γ
q
g.
12
A key parameter to
be estimated from the data is γ
q
. Since the inverse of q denotes the relative
price of equipment in terms of consumption goods, this quantity is determined
as the ratio of Gordon’s [104] price index of quality-adjusted equipment and
the price index for consumption. The parameters that are determined using
a priori information include the average growth rate of the investment-speciﬁc
shock γ
q
, the depreciation rate on structures δ
s
, and the tax rate on wage
income τ
l
. The ﬁrst parameter is determined using data on the inverse of the
relative price of investment goods following Gordon [104], and it is set at
12
From the production function, we have that g = γ
z
(γ
q
g)
α
e
g
α
s
, or
g = γ
1/(1−α
e
−α
s
)
z
γ
α
e
/(1−α
e
−α
s
)
q
,
g
e
= γ
1/(1−α
e
−α
s
)
z
γ
(1−α
s
)/(1−α
e
−α
s
)
q
.
October 9, 2009 15:12 9in x 6in b808-ch03
62 Business Cycles: Fact, Fallacy and Fantasy
1.032. Likewise, δ
s
= 0.056 and τ
l
= 0.40. The second set of parameters is
calculated using information on the long-run behavior of a set of the model’s
variables. This standard calibration exercise yields θ = 0.40, α
e
= 0.18,
α
s
= 0.12, ω = 1.59, τ
k
= 0.53, β = 0.97, g = 1.0124, and bh
ω
= 0.20,
where the parameters b and h cannot be identiﬁed separately. The restriction
that adjustment costs are zero along the balanced growth path yields the values
κ
e
= κ
s
= g. Furthermore, the symmetry condition φ
e
= (g/g
e
)
2
φ
s
≡ φ
leaves only the parameter φ to be determined.
The model is simulated to understand the contribution of the investment-
speciﬁc technological shocks in generating cyclical ﬂuctuations. Hence, only
the q shock is assumed to be operative. Unlike the standard technology shock
in RBC models, the investment-speciﬁc shock affects uses of factors such as
labor and capital through changed investment opportunities. A positive shock
to this variable works by increasing the rate of return to equipment investment,
which tends to raise the stock of equipment next period. Simultaneously, there
is a decline in the equipment’s replacement value, which raises utilization of
equipment today. This leads to increased employment and output. Given
that equipment investment is only 7% of GNP and 18% of the value of
output being derived from the use of equipment, the fraction of output
directly affected by the shock will typically be small. Hence, the impact
of the investment-speciﬁc technological shock depends on the quantitative
importance of the transmission mechanism. For this purpose, Greenwood
et al. [107] calibrate the adjustment cost parameter under two alternative
assumptions. According to the ﬁrst, the parameter φ is set to equalize the
consumption/output correlation implied by the model to that in the data.
This provides a lower bound on the contribution of the q shocks because
a positive shock to q reduces consumption at the same time as it increases
investment. A high value of φ works to reduce investment and to increase the
procyclicality of consumption. By contrast, a low value of the adjustment cost
parameter φ is chosen to make the volatility of investment in the model equal
to that in the data. This constitutes an upper value for the contribution of
the shocks q. They ﬁnd that under a high value of φ, the investment-speciﬁc
technological shocks explain around 28% of business cycle ﬂuctuations as
captured by the standard deviation of output. For a low value of φ, this ﬁgure
is 32%. They conclude that though investment-speciﬁc shocks contribute to
business cycle ﬂuctuations, they are not the main factor.
October 9, 2009 15:12 9in x 6in b808-ch03
Models of Business Cycles 63
3.5.2. Energy Shocks
From the beginning, the source of large supply-side shocks for the RBC
agenda has been sought in oil shocks. Barsky and Kilian [28] analyze the
relationship between oil shocks and changes in economic activity. Rotemberg
and Woodford [183] provide a rationale for the role of energy price shocks
in a model with imperfect competition. By contrast, Finn [92] argues
that it is possible to rationalize the role of energy shocks in models with
perfect competition. In her model, capital utilization is the channel through
which energy shocks enter the model. We brieﬂy describe her model before
concluding this chapter.
As in the standard RBC framework, there is a representative consumer
with preferences:
E
0
_
∞

t =0
β
t
u(c
t
, l
t
)
_
, 0 < β < 1,
with
u(c
t
, l
t
) =
_
c
α
t
(1 − l
t
)
1−α
_
1−σ
− 1
1 − σ
, 0 < α < 1, σ > 0.
The production function is given by
y
t
= (z
t
l
t
)
θ
(k
t
u
t
)
1−θ
, 0 < θ < 1,
where z
t
is an exogenous technology shock, k
t
is the stock of capital, and u
t
is
the rate of utilization of capital. As in the previous model, the depreciation of
physical capital depends on its utilization rate so that
k
t +1
= (1 − δ(u
t
))k
t
+ i
t
, δ(u
t
) =
ω
0
u
ω
1
t
ω
1
,
where 0 < δ(u
t
) < 1, ω
0
> 0, and ω
1
> 1. The new feature in Finn’s analysis
is that capital utilization requires energy:
e
t
k
t
=
ν
0
u
ν
1
t
ν
1
, ν
0
> 0, ν
1
> 1.
October 9, 2009 15:12 9in x 6in b808-ch03
64 Business Cycles: Fact, Fallacy and Fantasy
We can solve this equation for the utilization rate u
t
and substitute it into the
production function to obtain
y
t
= (z
t
l
t
)
θ
_
k
1−
1
ν
1
t
e
1
ν
1
t
_
ν
1
ν
0
_
1
ν
1
_
1−θ
.
Thus, the production of output depends on labor, capital, and energy. This
shows the direct effect of energy, but there is also an indirect effect which works
through the effect of the utilization rate on the depreciation of capital. Hence,
there are two channels for the transmission of energy shocks in the model.
Finally, the economy-wide resource constraint is given by
c
t
+ i
t
+ p
t
e
t
≤ y
t
,
where p
t
is the price of energy.
Finn [92] describes qualitatively and quantitatively the impact of increases
in the price of energy. First, an increase in p
t
reduces e
t
and u
t
through the
production function. The decrease in e
t
further reduces the marginal product
of labor, and hence the real wage w
t
and work effort l
t
. Thus, a positive energy
price shock operates in a similar way as a negative technology shock. If the price
shock is persistent, then the decrease in the values of e
t
, u
t
, and l
t
continue
into the future and reduce the future marginal productivity of capital. This
tends to reduce the investment i
t
and the future capital stock k
t
. The indirect
effect of the increase in the price of energy on capital’s future marginal energy
cost also tends to reduce investment and the future capital stock. Finn [92]
shows that the quantitative response of value-added and real wages to a shock
in energy prices implied by the model is in line with that in the data.
October 9, 2009 15:12 9in x 6in b808-ch04
Chapter 4
International Business Cycles
The international business cycle literature provides an extension to the
original real business cycle (RBC) approach by allowing for open-economy
considerations. One of the aims of the RBC approach has been to determine
whether two-country versions of the basic model can account for both the
co-movements studied in closed-economy models as well as international
co-movements, including correlations of macroeconomic aggregates across
countries and movements in the balance of trade. The closed-economy model
cannot be used to understand the propagation of shocks across countries or
regions, nor can it be used to discuss notions of international risk sharing.
Unlike the closed-economy model, open-economy business cycle models allow
for the role of technology processes in different countries in generating cyclical
ﬂuctuations. They can be used to model the impact of the ability to trade in
goods and to borrow and lend internationally. In this chapter, we review some
of the facts of international business cycles and discuss models that have been
used to rationalize the ﬁndings.
4.1. FACTS
One of the important areas of research has been to derive the so-called
stylized facts of international business cycles. Backus, Kehoe, and Kydland [25]
consider data on12developedcountries over a sample perioddating from1960
to 1990. They consider a two-country RBC model that has the features of the
original Kydland–Prescott model and that assumes a single homogeneous good
andcomplete contingent claims for allocating risk. They study the properties of
a baseline model against the actual features in the data. Backus et al. [26] study
65
October 9, 2009 15:12 9in x 6in b808-ch04
66 Business Cycles: Fact, Fallacy and Fantasy
the properties of the same model for ten developed countries plus a European
aggregate developed by the OECD between 1970 and 1990. The individual
countries they consider are Australia, Austria, Canada, France, Germany, Italy,
Japan, Switzerland, the United Kingdom, and the United States. The data are
ﬁltered using the Hodrick–Prescott ﬁlter. Among the important features of
their analysis is the estimation of the correlation properties of international
productivity shocks using data on estimated Solowresiduals. Thus, the exercise
in this literature is to replicate the correlation properties among the different
series given the variability and correlation properties of the shocks. In this
regard, Backus et al. [25, 26] estimate a bivariate AR(1) process for the domestic
and foreign shocks as
_
z
t
z
∗
t
_
= A
_
z
t −1
z
∗
t −1
_
+
_

t

∗
t
_
,
where z
t
, z
∗
t
denote the technology shocks to the domestic and foreign
countries, respectively, and
t
,
∗
t
denote the respective innovations to these
shocks. Based on several estimated speciﬁcations for the US and the European
aggregate, these authors use a symmetric speciﬁcation with
A =
_
0.906 0.088
0.088 0.906
_
,
with Std(
t
) = Std(
∗
t
) = 0.00852 and Corr(
t
,
∗
t
) = 0.258. The positive
off-diagonal entries allow for spillover effects of the shock to productivity in
one country to have a positive effect on the productivity of the other country.
The large autoregressive coefﬁcients displayed on the diagonal capture the
persistence in observed productivity.
The ﬁndings can be summarized under two headings:
• The quantity anomaly: In the data, the cross-country correlations of
output are greater than those of productivity, measured as the Solow
residual, and the cross-correlations of productivity are greater than those
of consumption. By contrast, the theoretical model implies the reverse
ranking.
• The price anomaly: The volatility of the terms of trade, deﬁned as the
relative price of imports to exports, is much higher in the data than it is
in the theoretical model.
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 67
Ambler, Cardia, and Zimmermann [14] extend the Backus et al. [26] sample
to 20 industrialized countries over the period 1960–2004. They estimate
the moments of interest using the generalized method of moments (GMM)
estimation, which allows them to obtain standard errors. The ﬁndings of
Ambler et al. [14] conﬁrm the ﬁndings of Backus et al. [25, 26], though
with lower magnitudes. They also ﬁnd that the cross-country correlations of
output, investment, and employment are positive and generally high in the
data; whereas in the model, they are negative.
The international RBCmodel produces these results because it implies that
the incentive is to use inputs where they are most productive. This fact leads
to negative cross-correlations between output, investment, and employment
in the theoretical model. In other words, suppose a positive technology shock
occurs in one country, say, the US. This leads to a strong investment response
in the US, which is accompanied by output and employment increases. Since
investment responds strongly to a positive productivity shock, the increase in
investment plus consumption is greater than the increase in output, which
leads to a trade deﬁcit in the domestic country. In a model with perfect capital
mobility, we thus observe a ﬂow of factors from other countries or regions
to the US. By contrast, what is typically observed in the data are positive co-
movements in these series across countries. Furthermore, investment and the
trade balance are more volatile in the model than they are in the data. The
two-country frictionless international business cycle model also implies that
consumers in different countries can share risk perfectly. This feature of the
model leads to the cross-country correlations of consumption being much
higher in the model than in the data. The price anomaly arises because the real
exchange rate in the baseline international RBC model is closely related to the
ratio of consumption across the two countries. With perfect risk sharing, this
ratio displays little volatility, implying that the real exchange rate is also less
volatile in the model than it is in the data.
In their original analysis, Backus et al. [25] also examine a variety of
perturbations of their original setup. First, they consider asymmetric spillovers
between the US and the European aggregate in which the response of US
productivity to shocks to European productivity is smaller than the response of
European productivity to shocks to US productivity. This changes the output-
investment correlation from positive to negative in the domestic country,
but the variability of investment and the trade balance continue to remain
October 9, 2009 15:12 9in x 6in b808-ch04
68 Business Cycles: Fact, Fallacy and Fantasy
counterfactually high, as does the positive correlation of consumption across
countries. Second, they consider

large spillovers across countries by changing
the off-diagonal elements of the A matrix from0.088 to 0.2 and the correlation
of the innovations from 0.258 to 0.5. With this change, they ﬁnd that
investment and the trade balance become much less volatile, and the cross-
country correlation of output goes from being negative to positive. However,
the cross-country correlation of consumption increases further. They also
examine various sorts of trading frictions. The ﬁrst of these involves introducing
a small trading friction in the form of a transport cost. They ﬁnd that this
friction has the effect of reducing the volatility of investment and the trade
balance, but has little effect on the cross-country correlations of consumption
and output. Next, they eliminate all trade in goods and assets by considering
an autarky situation. In this case, it is the correlation between technology
shocks that leads to any connection between countries. They ﬁnd that the
results are very similar to the case with a small transport cost. Surprisingly,
elimination of trade in state-contingent claims does not help to lower the
cross-country correlations of consumption. Backus et al. [25] argue that this
result is not due to international risk sharing considerations, but stems from
the permanent income hypothesis: when domestic and foreign productivity
shocks are correlated, a rise inproductivity inthe domestic country signals a rise
in productivity in the foreign country through the spillover effects. Hence, the
foreign agent increases consumption and reduces investment in anticipation
of future income increases.
4.2. THE ROLE OF INTERNATIONAL RISK SHARING
Before we continue with other extensions that have been proposed in the
international RBC literature, it is important to understand the role of
alternative ﬁnancial arrangements in generating the cross-country correlations.
Cole [71] argued that it is important to examine how alternative ﬁnancial
arrangements interact with the preferences and the production possibilities
set to determine the behavior of the real variables. The standard international
RBC model studied by Backus et al. [25, 26] considers the case with complete
contingent claims and perfect risk sharing across different countries. Cole [71]
and Cole and Obstfeld [72] examine a two-country real version of the Lucas
asset pricing model for this purpose (see Lucas [152]). Agents from both
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 69
countries are identical in terms of preferences and differ only in terms of
endowments. There are two goods, Y
1
and Y
2
. Country 1 has a random
endowment of good Y
1
and country 2 has a random endowment of good
Y
2
. Neither good is storable. We assume that endowments are stationary in
levels.
Let s
t
∈ S ⊆ R
m
+
denote a vector of exogenous shocks that follows a
discrete ﬁrst-order Markov process with a stationary probability transition
matrix P with (i, j) element p
i,j
, where p
i,j
= p(s
t
= s
j
|s
t −1
= s
i
). Let ˆ π
denote the vector of stationary probabilities which satisfy ˆ π = P

s
t +1
∈S
π(s
t +1
|s
t
)V
_
s
t +1
, z
j
1,t +1
, z
j
2,t +1
_
_
_
_
subject to the budget constraint (2.11) (or (2.12)). Let λ
i
(s
t
) denote the
Lagrange multiplier for country i in state s
t
. The ﬁrst-order conditions with
October 9, 2009 15:12 9in x 6in b808-ch04
74 Business Cycles: Fact, Fallacy and Fantasy
envelope conditions substituted in are given by
U
1
(c
1
1,t
, c
1
2,t
) = λ
1
(s
t
),
U
2
(c
1
1,t
, c
1
2,t
) = p(s
t
)λ
1
(s
t
),
U
1
(c
2
1,t
, c
2
2,t
) =
λ
2
(s
t
)
p(s
t
)
,
U
2
(c
2
1,t
, c
2
2,t
) = λ
2
(s
t
),
λ
i
(s
t
)q(s
t +1
, s
t
) = βπ(s
t +1
|s
t
)λ
i
(s
t +1
), i = 1, 2.
Simplifying these conditions yields
U
2
(c
i
1,t
, c
i
2,t
)
U
1
(c
i
1,t
, c
i
2,t
)
= p(s
t
), i = 1, 2 (2.13)
and
βπ(s
t +1
|s
t
)U
1
(c
i
1,t +1
, c
i
2,t +1
)
U
1
(c
i
1,t
, c
i
2,t
)
= q(s
t +1
, s
t
), i = 1, 2. (2.14)
Thus, we see that the Pareto optimal allocations can be supported in a complete
contingent claims equilibrium if
λ
i
(s
t
) =
µ
i
(s
t
)
φ
i
, i = 1, 2, where p(s
t
) =
µ
2
(s
t
)
µ
1
(s
t
)
.
In this equilibrium, we ﬁnd that consumers in each country equate their
marginal rates of substitution for each good across all possible current states.
If preferences satisfy the Cobb–Douglas assumption, then consumers in
each country consume a constant fraction of current output as speciﬁed
in equations (2.7)–(2.10). This follows from the fact that the competitive
equilibrium is Pareto optimal. Hence,
p(s
t
) =
(1 − α)c
i
1,t
αc
i
2,t
=
(1 − α)y
1,t
αy
2,t
.
In this case, the contingent claims contracts are considered to be consistent
with the results on consumption for each country. Given the solution for
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 75
the relative price p(s
t
) and the consumption allocations given in equations
(2.7)–(2.10), notice that the solution
z
i
1
(s
t +1
) = z
i
2
(s
t +1
) = 0, z
i
1,t
= z
i
2,t
= 0
with δ = α satisﬁes the consumers’ budget constraints and the market-clearing
conditions. But this is a contingent claims equilibrium in which no assets are
traded! In such an equilibrium, it is still possible to price the contingent
claims using the expression in (2.14). In the next section, we will demonstrate
explicitly that the Pareto optimal allocations can indeed be attained in a no-
asset-trading equilibrium. Since the price of any ﬁnancial asset can be obtained
as a function of the contingent claims prices, it follows that any ﬁnancial asset
can also be priced in such an equilibrium.
4.2.3. No Asset Trading
Now suppose that there is trade in goods but no trade in international assets.
In this case, we are forcing the current account to equal zero in each period.
Country 1 and 2’s budget constraints can be expressed, respectively, as
c
1
1,t
+ p
t
c
1
2,t
= y
1,t
, (2.15)
c
2
1,t
p
t
+ c
2
2,t
= y
2,t
. (2.16)
Once again, this is a static problembecause there are no assets for intertemporal
consumption smoothing and the endowment is nonstorable. Let µ
j
t
denote
the Lagrange multiplier on the budget constraint of country j. If we assume
the Cobb–Douglas functional form for preferences, then the consumption
allocations are
c
1
1,t
= αy
1,t
,
c
1
2,t
=
(1 − α)y
1,t
p
t
,
c
2
1,t
= αp
t
y
2,t
,
c
2
2,t
= (1 − α)y
2,t
.
October 9, 2009 15:12 9in x 6in b808-ch04
76 Business Cycles: Fact, Fallacy and Fantasy
Equilibrium in the two goods markets requires that
c
1
1,t
+ c
2
1,t
= y
1,t
, (2.17)
c
1
2,t
+ c
2
2,t
= y
2,t
. (2.18)
Substitute the consumption functions into the market-clearing conditions and
solve for the relative price to show that
p
t
=
(1 − α)y
1,t
αy
2,t
.
This price can be substituted into the consumption functions above to
show that
c
1
2,t
= αy
2,t
,
c
2
1,t
= (1 − α)y
1,t
.
Notice that the no-asset-trading allocation is identical to the central planning
allocation if
α = δ =
_
1 +
_
φ
2
φ
1
_
σ
_
,
or
φ
1
=
_
1 +
_
1 − α
α
_
ρ
_
−1
and φ
2
= 1 − φ
1
.
This exercise illustrates several points:
• The absence of international capital mobility does not necessarily imply
that the allocation is not Pareto optimal. Efﬁcient risk sharing can occur
despite the lack of ﬁnancial assets and insurance.
• The international ratio of marginal utilities across countries is identical
across goods and states. A large and positive shock in the amount of good
y
1,t
is positively transmitted to the residents of country 2 by the increase in
demand (and hence the relative price) for good y
2,t
. Alarge negative shock
in the amount of a good is similarly transmitted across borders, despite
the absence of trade in ﬁnancial assets. Hence, efﬁcient risk sharing occurs
through changes in the relative price of goods.
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 77
• Notice that we can price ﬁnancial assets in the model, under the
assumption that the current account is zero, and can showthat real interest
rates and real asset returns will be equal for the two countries, despite the
absence of ﬁnancial capital mobility.
• The total consumption of countries 1 and 2 is
c
1
1,t
+ p
t
c
1
2,t
= αy
1,t
+ αy
2,t
,
c
2
1,t
+ p
t
c
2
2,t
= (1 − α)y
1,t
+ (1 − α)y
2,t
.
Notice that correlation of the total value of consumption of country 1
and country 2 is positive and equal to one.
As Cole and Obstfeld [72] point out, the positive transmission of shocks occurs
because countries specialize in the production of goods.
4.2.4. Nonspecialization in Endowments
To illustrate the impact of nonspecialization, Cole and Obstfeld [72] introduce
a third good. Call this third good w. Assume that both countries receive an
exogenous and stochastic endowment of good w and let w
j
: S → W =
[w, ¯ w] denote the realization of good w in country j. Let α
1
, α
2
, α
w
denote
the expenditure shares under the assumption of Cobb–Douglas preferences
and let w be the numéraire good, so that p
1,t
denotes the relative price of good
y
1,t
in terms of w
t
and p
2,t
denotes the relative price of good y
2,t
in units of
w
t
. Agents in countries 1 and 2 have budget constraints given, respectively, by
p
1,t
c
1
1,t
+ p
2,t
c
1
2,t
+ c
1
w,t
≤ p
1,t
y
1,t
+ w
a
t
,
p
1,t
c
2
1,t
+ p
2,t
c
2
2,t
+ c
2
w,t
≤ p
2,t
y
2,t
+ w
b
t
.
The equilibrium relative prices satisfy
p
1,t
=
α
1
[w
1
t
+ w
2
t
]
α
w
y
1,t
,
p
2,t
=
α
2
[w
1
t
+ w
2
t
]
α
w
y
2,t
.
October 9, 2009 15:12 9in x 6in b808-ch04
78 Business Cycles: Fact, Fallacy and Fantasy
The consumption of good 1 by agents in countries 1 and 2, under the
assumption of Cobb–Douglas preferences, can be shown to satisfy
c
1
1,t
=
_
α
w
_
w
1
t
w
1
t
+ w
2
t
_
+ α
1
_
y
1,t
,
c
2
1,t
=
_
α
w
_
w
2
t
w
1
t
+ w
2
t
_
+ α
2
_
y
1,t
.
Similar expressions can be derived for the consumption of goods y
2
and w.
The ratio of marginal utilities of both countries for each good will be equal
to a constant across all states, a condition for Pareto optimality, only if the
share of the endowment of w
t
, (
w
1
t
w
1
t
+w
2
t
) and (
w
2
t
w
1
t
+w
2
t
), is constant as the total
w
t
varies with s
t
. The shocks to w
1
t
and w
2
t
must be perfectly correlated for
the allocation with no trade in ﬁnancial assets to be Pareto optimal. If these
shocks are not perfectly correlated, then it is beneﬁcial to trade equity shares
or other forms of ﬁnancial assets.
This helps to clearly distinguish between country-speciﬁc shocks, which
affect all sectors within a country, and industry-speciﬁc shocks. Shocks to y
1,t
or y
2,t
are, by deﬁnition, country-speciﬁc shocks; whereas shocks to w
1
t
, w
2
t
,
where w
1
t
, w
2
t
are not perfectly correlated, are sector-speciﬁc shocks. Hence,
when there are sector-speciﬁc shocks, there are gains to asset trading that
improve risk sharing and allow diversiﬁcation. The intuition is that, in the
absence of trade in ﬁnancial assets, the country with a negative shock to the
endowment of w
t
would like to run a current account deﬁcit by importing
w
t
and borrowing against future endowment. Since the current account must
always be balanced, the country must export more of the good in which it
specializes in production to ﬁnance the import of good w
t
. Notice that the
relative price of w
t
may not adjust much if w
1
t
and w
2
t
are negatively correlated
but the sum w
1
t
+ w
2
t
ﬂuctuates very little.
4.2.5. Nontraded Goods
Suppose now that the third good is a nontraded good. Call this good n and
assume that n
j
: S → N = [n, ¯ n] denotes the realization of good n in
country j. Let α
1
, α
2
, α
n
denote the expenditure shares under the assumption
of Cobb–Douglas preferences and let y
1
be the numéraire good. Under the
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 79
assumption of balanced trade and Cobb–Douglas preferences, the demands
for the goods satisfy
c
1
1,t
=
α
1
1 − α
n
y
1,t
,
c
1
2,t
=
α
1
1 − α
n
y
2,t
,
c
2
1,t
=
α
2
1 − α
n
y
1,t
,
c
2
2,t
=
α
2
1 − α
n
y
2,t
.
Each country consumes its endowment of the nontraded good, n
1
t
, n
2
t
. The
ratio of marginal utility across countries for a traded good will now depend
on the ratio
n
1
t
n
2
t
. Unless n
1
t
, n
2
t
are perfectly correlated, then the resulting
allocations will not be Pareto optimal. There are gains from international risk
sharing through asset trade. Notice that the correlation of consumption across
countries will now depend on the proportion of a country’s consumption that
is nontradeable. If this sector constitutes a large fraction of consumption, then
even if consumption of traded goods is perfectly correlated, the correlation
of national consumption levels may be close to zero. Thus, nontraded goods
provide one vehicle for reducing the large cross-country correlations implied
by the baseline model.
4.2.6. Trade in Equity Shares
We nowintroduce trade in ﬁnancial assets. An agent in country 1 holds equity
shares that are claims to the endowment stream for good y
1
(the domestic
good) and claims to y
2
(the foreign good). We now discuss the impact of asset
trading and relate it to the model without asset trade discussed earlier.
Let z
j
i,t
for j = 1, 2 and i = 1, 2 denote the shares of good i held
by an agent in country j at the beginning of period t . An agent’s budget
constraint is
z
j
1,t
[y
1,t
+ q
1,t
] + z
j
2,t
[p
2,t
y
2,t
+ q
j
2,t
]
≥ c
j
1,t
+ p
2,t
c
j
2,t
+ q
1,t
z
j
1,t +1
+ q
2,t
z
j
2,t +1
. (2.19)
October 9, 2009 15:12 9in x 6in b808-ch04
80 Business Cycles: Fact, Fallacy and Fantasy
Let µ
j
t
denote the Lagrange multiplier. The agent maximizes his objective
function subject to the constraint. The ﬁrst-order conditions are
U
1
(c
j
1,t
, c
j
2,t
) = U
2
(c
j
1,t
, c
j
2,t
)p
2,t
, (2.20)
U
1
(c
j
1,t
, c
j
2,t
)q
1,t
= βE
t
U
1
(c
j
1,t +1
, c
j
2,t +1
)[q
1,t +1
+ y
1,t +1
], (2.21)
U
1
(c
j
1,t
, c
j
2,t
)q
2,t
= βE
t
U
1
(c
j
1,t +1
, c
j
2,t +1
)[q
2,t +1
+ p
2,t +1
y
2,t +1
]. (2.22)
In equilibrium, all equity shares are held and the endowment of each good
is completely consumed. Lucas [152] assumes that agents hold identical
portfolios, so that z
j
i,t
= 1/2 for j = 1, 2 and i = 1, 2 so that φ
j
= 1/2
and δ = 1/2. In such a world, national wealth is equal across countries
and agents have perfectly diversiﬁed portfolios. Agents across countries have
identical consumption in this case, unlike the economy in which there is no
trade in ﬁnancial assets. In the initial model described earlier, there was no
trade in ﬁnancial assets and specialization in endowments and the allocation
was Pareto optimal. We commented that we could price ﬁnancial assets even
if these assets were not traded. If we assume that z
1
1,t
= z
1
2,t
= α and
z
2
1,t
= z
2
2,t
= 1 − α, then the equilibrium allocation with no trade in
ﬁnancial assets can be achieved. Hence, we can achieve at least two stationary
allocations, depending on the initial distribution of the claims. This simple
example, when combined with our discussion of the equilibriumwith no trade
in ﬁnancial assets, illustrates an important point. International risk sharing can
be achieved through ﬂuctuations in relative prices in the current account and
by trade in ﬁnancial assets. In particular, it does not require the existence
of a full set of contingent claims. The presence of nontraded goods or lack
of specialization in production of a good can affect how much consumption
insurance can be achieved through relative price ﬂuctuations. If we introduced
trade in equity shares when there is a third good w that is produced by
both countries, then portfolio diversiﬁcation may require that an agent hold
equity shares for w
1
and w
2
if the endowment shocks for w are not perfectly
correlated. If these shocks are perfectly correlated, thenportfolio diversiﬁcation
may be achieved by specializing in the holding of equity shares of one
country only.
There has been substantial literature on the lack of international portfolio
diversiﬁcation and the degree of international consumption risk sharing. The
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 81
international portfolio diversiﬁcation puzzle is the notion that investors hold
too little of their wealth in foreign securities to be consistent with the standard
theory of portfolio choice. Baxter and Jermann [36] argue that the failure of
international diversiﬁcation is substantial. Their model incorporates human
and physical capital and, within the context of their model, optimal behavior
would lead to a short position in domestic assets because of a strong positive
correlation between the returns to human and physical capital. A more recent
paper by Heathcote and Perri [117] extends the Baxter and Jermann model to
include more than one traded good. They ﬁnd, as we have noted above, that
consumption insurance is available through relative price ﬂuctuations and that
these price ﬂuctuations are capable of achieving efﬁcient risk sharing. Clearly,
the conclusion on whether there is sufﬁcient or insufﬁcient risk sharing is very
sensitive to model speciﬁcations.
Empirical evidence on international consumption risk sharing is provided
in Backus et al. [25], who show that the data are inconsistent with the
implications for consumption for the baseline international RBC model. By
contrast, Devereux, Gregory, and Smith [81] showthat preferences that display
a nonseparability between consumption and leisure can help to reduce the
consumption correlations implied by the model. Lewis [145] also documents
that there is insufﬁcient intertemporal risk sharing in consumption. As we have
noted above, the existence of nontraded goods, combined with the assumption
that utility is nonseparable in traded and nontraded goods, makes it more
difﬁcult to determine the optimal degree of consumption risk sharing. We
have also shown above that relative price ﬂuctuations can be a substitute for
trade in ﬁnancial assets in achieving consumption insurance. Lewis [145]
documents that the nonseparability of utility or the restriction of asset trade
alone is not enough to explain the risk sharing that we observe, but that when
nonseparability and asset trade restrictions are combined, she cannot reject the
hypothesis that there is risk sharing.
4.2.7. Limited Risk Sharing
As described earlier, whether there exists perfect risk sharing in the data appears
to depend on the model speciﬁcation adopted by the researcher. Nevertheless,
one could ask whether limited risk sharing opportunities among consumers
in different countries could help to better reconcile the data with the model.
October 9, 2009 15:12 9in x 6in b808-ch04
82 Business Cycles: Fact, Fallacy and Fantasy
In this vein, credit market frictions and restrictions on international capital
ﬂows constitute a proposed channel for the propagation of international
shocks. Kollman [136] and Baxter and Crucini [34] examine models where
only non-contingent bonds can be traded internationally. These authors ﬁnd
that incomplete asset markets help to reduce the cross-country correlation
of consumption, but the cross-country correlations of output, investment,
and hours worked remain counterfactually negative. Moreover, the results for
the cross-country correlations are obtained only if the productivity shocks
are highly persistent and there are very little spillover effects across countries.
Heathcote and Perri [117] examine the implications of a two-country model
under alternative assumptions about the ﬁnancial structure, given a trade
structure. They show that the model matches the correlations in the data
under a ﬁnancial autarky assumption. Their analysis involves extending the
analysis in Cole and Obstfeld [72] to incorporate an explicit production side.
Kehoe and Perri [126] consider a model where market incompleteness is
obtained endogenously through the introduction of imperfectly enforceable
international loans. In their framework, a country can incur international
indebtedness only to the extent that such indebtedness can be enforced
through the threat of exclusion fromfuture intertemporal and interstate trade.
This analysis builds on the earlier works of Kehoe and Levine [124, 125],
Kocherlakota [135], Alvarez andJermann[12], andothers ondebt-constrained
asset markets. In these models, the inability of agents to share risk perfectly and
to fully offset the effects of idiosyncratic shocks leads to lower cross-country
correlations of consumption and higher cross-country correlations of output.
Kehoe and Perri [126] consider a standard international RBC model with
production and capital accumulation. Output in each country is produced
using capital and labor according to the constant-returns-to-scale production
function:
F(k
i
(s
t −1
), A
i
(s
t
)l
i
(s
t
)),
where A
i
(s
t
) is a randomshock. The preferences of the representative consumer
in each country are given by
∞

i=1
_
F(k
i
(s
t −1
), A
i
(s
t
)l
i
(s
t
)) + (1 − δ)k
i
(s
t −1
)
_
. (2.24)
The innovation in Kehoe and Perri [126] is to formulate a version of
this problem that allows for enforcement constraints. These require that each
country prefers the allocation it receives to the one that it could attain if it
were in (ﬁnancial) autarky from then onwards. The enforcement constraints
are expressed as
∞

i
β
t
π(s
t
)
_
M
i
(s
t −1
)U(c
i
(s
t
), l
i
(s
t
))
+µ
i
(s
t
)[U(c
i
(s
t
), l
i
(s
t
)) − V
i
(k
i
(s
t −1
), s
t
)]
_
plus the terms relating to the resource constraints. In this expression, M
i
(s
t −1
)
satisﬁes
M
i
(s
t
) = M
i
(s
t −1
) + µ
i
(s
t
), t ≥ 0
with M
i
(s
t −1
) = φ
i
. Thus, M
i
(s
t
) are just the original planning weights plus
the sum of the multipliers µ
i
(s
t
) along the path s
t
.
It is beyond the scope of this book to derive a solution for the model
with enforcement constraints. Therefore, we will focus on the results of
simulating this model and compare it with alternatives that have been
obtained in the literature. Kehoe and Perri [126] generate solutions for three
different economies: complete markets, a bondeconomy, andaneconomy with
enforcement constraints. The bond economy stipulates that all intertemporal
trades must be implemented with an uncontingent bond. Thus, the budget
constraints for households in each country are expressed as
c
i
(s
t
) + k
i
(s
t
) + q(s
t
)b
i
(s
t
) ≤ w
i
(s
t
)l
i
(s
t
)
+[r
i
(s
t
) + (1 − δ)]k
i
(s
t −1
) + b
i
(s
t −1
),
2
For other applications, see Marcet and Marimon [155].
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 85
where w
i
(s
t
) and r
i
(s
t
) denote the wage rate and the rental rate on capital in
country i, q(s
t
) is the period t price of an uncontingent that pays off one unit
in period t +1 regardless of the state, and b
i
(s
t
) is the quantity of the bond by
consumers in country i. There is also a borrowing constraint that requires that
borrowing cannot exceed some ﬁnite bound, b(s
t
) ≥ −
¯
b, where 0 <
¯
b < ∞.
Preferences in each country are taken to be of the form
U(c, l ) =
_
c
γ
(1 − l )
1−γ
_
1−σ
1 − σ
, 0 < γ < 1, σ ≥ 0, (2.26)
and the production function as
F(k, AL) = k
α
(Al )
1−α
, 0 < α < 1. (2.27)
The parameter values that are considered are standard, and given by β =
0.99, γ = 0.36, σ = 2, α = 0.36, and δ = 0.025. The coefﬁcients of
the A matrix, which governs the persistence properties of the shocks and the
spillovers between them, are parameterized in different ways. According to one
parameterization intended to capture substantial persistence, the autoregressive
coefﬁcients of the shocks for each country captured by the diagonal elements
of the A matrix are set at 0.9 and the off-diagonal elements are set at zero. These
values are consistent with those assumed by Kollman [136] and Baxter and
Crucini [34]. A second parameterization allows for high persistence with the
diagonal elements equal to 0.99 and the off-diagonal elements equal to zero,
and a third one allows for high spillover with the diagonal elements equal to
0.85 and the off-diagonal elements equal to 0.15. Similar to Backus et al. [25],
the variance of the innovations to the productivity shocks in each country is
set at Var(
i,t
) = 0.007 and the correlation of the shocks is set at 0.25.
The ﬁndings from the model echo many of the earlier ﬁndings. Under
the complete markets speciﬁcation, the model displays many of the anomalies
reported in the literature. First, the consumption correlations are signiﬁcantly
higher than the output correlations in the model, whereas these correlations are
the opposite in the data. Second, the cross-country correlations of employment
and investment are negative in the model, whereas they are positive in the data.
Third, net exports and investment are much more volatile in the model than
they are in the data. In the bond economy, the three discrepancies between
the model and the data remain, albeit with some minor improvements in the
various statistics. In the economy with enforcement constraints, the output and
October 9, 2009 15:12 9in x 6in b808-ch04
86 Business Cycles: Fact, Fallacy and Fantasy
consumption correlations are both positive and closer to each other, although
the cross-country correlation of consumption is greater than that of output.
Second, the cross-country correlations of employment and investment have
now switched from being negative to positive; and third, the volatility of
net exports and investment has declined dramatically. The only remaining
discrepancy is that net exports are procyclical in the model whereas they are
countercyclical in the data.
As discussed earlier, the frictionless international RBC model implies that
investment ﬂows to the country with the higher productivity shock, leading
to very volatile investment and net exports. In the literature, exogenous
adjustment costs are typically added to inhibit the ﬂow of investment. Kehoe
and Perri [126] note that the enforcement constraint acts as an endogenous
inhibiting factor. Considering economies with exogenous adjustment costs to
investment, they ﬁnd that the volatilities of investment and net exports are
diminished, but that the anomalies regarding the cross-correlations of output,
consumption, investment, and employment remain. Finally, the authors
examine the response of the variables in domestic and foreign countries to
a positive shock to productivity in the domestic country. This increases the
productivity of both capital and labor in the domestic country, so resources are
optimally shifted there. The capital stock in the domestic country increases, as
domestic residents save more and also as investment from abroad ﬂows there.
The net ﬂow of investment increases the trade deﬁcit in the domestic country.
In the foreign country, we notice declines in employment and investment.
Thus, we see that the differing responses of the domestic versus the foreign
country lead to the negative cross-country correlations of employment and
investment. Since residents of each country can acquire contingent claims
that allow them to smooth consumption across all possible history of shocks,
risk sharing across countries implies that consumption increases in the foreign
country. Consumption increases substantially more in the domestic country
because consumptionandlabor complement eachother. Hence, we observe the
highcross-country correlations betweendomestic andforeignconsumption. In
the bond economy, the responses are similar to those for the complete markets
economy, but because of restrictions on asset trading, they are somewhat
dampened.
Finally, let us consider the impact of a positive productivity shock in the
enforcement economy. Suppose that the social planner tries to implement
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 87
the complete markets allocation for this economy. The high and persistent
productivity shock in the domestic country increases the value of autarky,
increasing the incentives for default. The increased investment ﬂows to the
domestic country further increase the value of autarky. Hence, the planner
must restrict investment to the domestic country to prevent the default option
from being exercised. Furthermore, the planner also builds up the capital
stock in the foreign country to ensure that the risk-sharing arrangement
between the domestic and foreign countries is not reneged upon. In terms
of the behavior of consumption, the complete markets allocation implies
that an increase in output in the domestic country will lead to increases in
consumption in both the domestic and foreign countries. In the presence of
an enforcement constraint, however, this is not possible. Hence, to ensure that
domestic country residents consume more than foreign country residents, the
social planner increases the relative weight on the utility of domestic country
residents (recall that the social planner’s weight is the inverse of the marginal
utility of consumption). Over time, as the productivity shock dies out, the
value of autarky diminishes and the relative weight on the home country also
falls.
4.3. OTHER EXTENSIONS
The international RBC literature has sought to reconcile the ﬁndings in the
data not only in terms of the cross-country consumption correlations, but also
for the behavior of output, investment, and the real exchange rate. As a result,
models that relax assumptions regarding preferences, the production side, and
the presence of additional shocks have been developed. We describe a few of
these extensions in what follows.
Hess and Shin [119] re-explore the two international business cycle
anomalies emphasized by Backus et al. [25] as well as establish the pattern
of productivity growth between industries and countries. They then compare
these ﬁndings for the international business cycle to those obtained for data
between regions within a country — the so-called “intranational business
cycle”. They argue that the intranational business cycle is a natural environment
for thinking about the interactions between economies when there are no
trade frictions and when there are not multiple currencies. Ambler et al.
[13] modify the supply side of a two-country model by adding multiple
October 9, 2009 15:12 9in x 6in b808-ch04
88 Business Cycles: Fact, Fallacy and Fantasy
sectors and trade in intermediate goods. The model generates a higher cross-
country correlation of output than standard one-sector models. It also predicts
cross-country correlations of employment and investment that are closer to
the data.
Stockman and Tesar [200] introduce a nontraded goods sector in each
country. Similar to our discussion above, they argue that introducing
nontraded goods may be a way of re-establishing the link between a
nation’s output and its spending. They employ preferences that depend on
consumption of the tradable goods of countries 1 and 2, a nontradable good,
and leisure. Preferences are assumed to be nonseparable with respect to a
composite good, consisting of the tradable goods of countries 1 and 2, and
a nontradable good. In their model, output in the traded and nontraded
goods sector of each country is produced using sector-speciﬁc capital and labor
which is mobile between sectors. However, there is no international capital
or labor mobility. The authors ﬁnd that the model predicts the correlation
between home and foreign output, overstates the cross-country correlation of
aggregate consumption, and greatly overstates the cross-country correlation
of tradable consumption. Hence, they ﬁnd that introducing nontradable
goods does not sufﬁce to reduce the cross-country correlation of consumption.
Likewise, they ﬁnd that the model overstates the negative correlation between
the relative price of nontradable to tradable goods and relative consumption of
nontraded to traded goods, and understates the variability of the trade balance.
They conclude that an international business cycle model driven solely by
productivity shocks cannot account for the ﬁndings. Instead, they argue that
what is needed is a source of nation-speciﬁc shocks that shift demand. They
introduce taste shocks that affect the utility of traded versus nontraded goods
and ﬁnd that this feature brings the data more in line with the implications of
the model.
Baxter and Farr [35] develop a model with variable capital utilization
as a way of accounting for the international correlations. They argue that
variable capacity utilization has the potential to account for the co-movement
of factor inputs across countries. We already considered the role of variable
factor utilization in reconciling the behavior of procyclical productivity in
closed-economy business cycle models (see also Burnside and Eichenbaum
[51], Basu [29], or Basu and Kimball [30]). The notion is that a positive shock
to productivity in the domestic country will tend to reduce the investment
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 89
ﬂowacross countries by leading to an increase in capital utilization rather than
investment. As we described above, many of the puzzles of the international
business cycle literature can be addressed successfully by devising a mechanism
to limit investment ﬂows across borders. Kehoe and Perri [126] achieve this
through an enforcement constraint which requires that the utility under the
constrained allocation exceed the utility that country could obtain under
autarky after defaulting on its international debt obligations.
Baxter and Farr [35] assume that preferences are given by the speciﬁcation
in equation (2.26). Output in the domestic country is produced using capital
services S
t
and labor L
t
as
Y
t
= A
t
S
1−α
t
(X
t
L
t
)
α
, 0 < α < 1,
where capital services are the product of the capital stock, K
t
, andthe utilization
rate, Z
t
:
S
t
= Z
t
K
t
.
The capital accumulation process displays both costly utilization of capital and
adjustment costs:
K
t +1
= [1 − δ(Z
t
)] + φ(I
t
/K
t
)K
t
,
where δ

> 0, δ

> 0, φ

> 0, and φ

< 0. Similar functions characterize the
foreign country. International trade in assets is made solely with one-period,
real pure discount bonds which have the price q
t
= [1 + r
t
]
−1
. Assuming
that the household in each country owns the capital stock and makes real
investment decisions, the budget constraint for the representative household
is given by
C
t
+ I
t
+ q
t
B
t +1
≤ Y
t
+ B
t
,
where B
t +1
is the quantity of bonds carried over into the next period.
The parameterization of the capital utilization function and the adjustment
cost function are key aspects of the quantitative analysis. The covariance
properties of the technology processes for the domestic and foreign technology
shocks are parameterized to be near unit roots with zero spillover effects. The
correlation of the innovations to the technology shocks is set at 0.258, which
is consistent with earlier studies. Finally, the variance of the innovations is
set so that the variance of output in the model exactly matches the volatility
October 9, 2009 15:12 9in x 6in b808-ch04
90 Business Cycles: Fact, Fallacy and Fantasy
of the US economy. One of the key ﬁndings regarding the role of variable
capital utilization is that, ﬁrst, the volatility of the shocks necessary to match
output volatility is substantially reduced. Second, introducing variable capital
utilization reduces the volatilities of consumption, capital, and exports, which
is desirable, and also those of hours and employment, which is not. The most
important impact of variable factor utilization is on the cross-correlations of
the factor inputs. Speciﬁcally, the cross-correlation of hours and investment
become positive. Intuitively, variable capital utilization takes the place of
investment ﬂows across countries. When a positive productivity shock occurs
in one country, capital utilization rates increase, which in turn increase
employment. Even with a modest cross-country correlation of the innovations,
this is accompanied by increases in investment and employment in the other
country.
4.4. PUZZLES REVISITED
The puzzles in the international business cycle literature have been the topic
of much research (see, for example, the survey by Baxter [33]). In their review,
Obstfeld and Rogoff [168] argue that many of the puzzles in international
macroeconomics may just be speciﬁc to the models researchers are using.
In our earlier discussion, we described the puzzles that arose when trying
to match the international RBC model to the data. Obstfeld and Rogoff
[168] provide a broader view of the empirical puzzles in the international
macroeconomics literature, though they also relate their discussion to the
international RBC literature. They enumerate the following discrepancies
between data and existing theory as “puzzles”:
• Home bias in trade: A number of authors have documented that intra-
national trade is typically much greater than international trade. See, for
example, McCallum [157] or Helliwell [118].
• The Feldstein–Horioka puzzle: According to this puzzle, long averages
of saving rates and investment rates tend to be correlated for the OECD
countries (see Feldstein and Horioka [89]). Yet, in a world of integrated
capital markets, we would expect capital to ﬂowto regions where the rates
of return are highest.
October 9, 2009 15:12 9in x 6in b808-ch04
International Business Cycles 91
• Home bias in equity portfolios: This reﬂects the puzzling preference of
stock market investors for home assets (for a recent elaboration, see Tesar
and Werner [203]). In Section 4.2, we discussed various explanations that
account for this puzzle — the presence of human capital as discussed by
Baxter and Jermann [36] or nontraded goods.
• The international consumption correlation puzzle: We already discussed
the ﬁnding that cross-country consumption correlations exceed the cross-
country output correlations ina typical international RBCmodel, whereas
the opposite is true in the data. Backus and Smith [24] show that in an
economy with traded and nontraded goods, perfect risk sharing across
countries implies that countries which experience declines in the relative
price of consumption should receive large transfers of goods.
• The purchasing power parity (PPP) puzzle: The PPP puzzle arises from
the fact that shocks to the real exchange rate are very persistent.
• The exchange rate disconnect puzzle: This puzzle captures the notionthat
there exists a relationship between exchange rates and any macroeconomic
variable. Meese and Rogoff [161] further show that most exchange rate
models forecast exchange rates no better than a naive random walk.
Surprisingly, Obstfeld and Rogoff [168] ﬁnd that adding transport costs goes
a long way towards accounting for the ﬁrst ﬁve puzzles. They argue that,
ﬁrst, the international consumption correlation puzzle tends to be speciﬁc
to a model’s assumptions, as we described above, and does not have the same
weight as, say, the international portfolio diversiﬁcationpuzzle. Second, the last
two puzzles are pricing puzzles, whereas the other four refer to the behavior
of quantities. They also note that the pricing puzzles require such features
as nominal rigidities of the type that we will consider in the next chapter.
In contrast to some of the other contributions that we have discussed, the
analysis of Obstfeld and Rogoff [168] is an attempt to unify the explanations
of a related but disparate set of ﬁndings. Clearly, the literature that we have
described has revealed a variety of promising directions for future research.
October 9, 2009 15:12 9in x 6in b808-ch04
This page intentionally left blank This page intentionally left blank
October 9, 2009 15:12 9in x 6in b808-ch05
Chapter 5
New Keynesian Models
The New Keynesian approach has gained signiﬁcance in the modern
macroeconomics literature. Critics of the original real business cycle (RBC)
approach had, from the onset, taken issue with the notion that prices can
adjust costlessly to clear markets. More recently, New Keynesian theories have
revived interest in business cycle models that are capable of producing short-
run economic ﬂuctuations based on the types of forces that Keynes had initially
postulated. Prototypical New Keynesian models such as that by Rotemberg
and Woodford [182] allow for imperfect competition and markups to capture
alternative propagation mechanisms in response to technology shocks or
shocks to government expenditures. Limited participation models also allow
alternative mechanisms for the propagation of real and monetary shocks.
1
The behavior of hours and productivity has been a topic of debate. The
RBC conclusions regarding the response of hours, output, and other variables
to a technology shock have been questioned by empirical results obtained
along several different lines. On the one hand, Gali [96] has argued that in a
suitably restricted vector autoregression (VAR) including measures of hours,
productivity, output, and other variables, the response of hours to productivity
shocks is negative. This is in contrast to the RBC model, which predicts
that hours rise on impact to a positive technology shock. Basu, Fernald, and
Kimball [32] also present evidence that hours worked and other variables fall
in response to technology improvements in the short run. Their approach
involves purging the standard Solow residual of factors that might lead to
procyclicality, suchas variable factor utilization. There alsoexist open-economy
versions of the New Keynesian model that have been used for policy analysis
1
For a review and discussion of these models, see Christiano, Eichenbaum, and Evans [66].
93
October 9, 2009 15:12 9in x 6in b808-ch05
94 Business Cycles: Fact, Fallacy and Fantasy
and for understanding the role of monetary shocks.
2
We ﬁrst describe a simple
New Keynesian framework that can be used to rationalize the observations,
and then discuss the empirical ﬁndings in more detail.
5.1. THE BASIC MODEL
Consider a simple New Keynesian model with monopolistic competition,
price rigidities, and variable labor effort due to Gali [96]. Suppose that a
representative household chooses consumption C
t
, money holdings M
t
, hours
worked N
t
, and effort levels U
t
to maximize
E
0
_
∞

m
t
_
= C(L)
t
,
where
z
t
and
m
t
denote the sequences of technology and non-technology
shocks, respectively. The orthogonality assumption, together with a
normalization, implies that E(
t

t
) = I . The identifying assumption is that
only technology shocks have a permanent effect on productivity, which can
be expressed as the restriction C
12
(1) = 0.
3
Gali [96] estimates this model
using postwar US data. Surprisingly, he ﬁnds that alternative measures of labor
input decline in response to a positive technology shock while GDP adjusts
only gradually to its long-run level. Furthermore, technology shocks explain
only a small fraction of employment and output ﬂuctuations. By contrast,
Gali [96] ﬁnds that variables that have no permanent effects on employment
(and which are referred to as demand shocks) explain a substantial fraction of
the variation in both employment and output. Christiano, Eichenbaum, and
Vigfusson [68] suggest that the standard RBC results hold if per capita hours
are measured in log-levels as opposed to differences.
Chari, Kehoe, and McGrattan [63] have challenged the ﬁndings derived
from so-called SVARs. First, they argue that the difference speciﬁcation for
aggregate hours is a priori misspeciﬁed because all RBC models imply that
per capita hours is a stationary variable. Second, they argue that if the
simulated data from a simple RBC model are subjected to a SVAR-based
test, the difference-stationary version of the model implies that the response
3
The value of the matrix C(L) evaluated at L = 1 gives the long-run multipliers for the model, i.e.,
the long-run impact of a given shock.
October 9, 2009 15:12 9in x 6in b808-ch05
100 Business Cycles: Fact, Fallacy and Fantasy
of hours to a shock to productivity is negative as in Gali and Rabanal [98].
However, if hours worked is considered to be stationary in levels, then they
argue that the conﬁdence bands for the impulse response functions from the
SVAR are so wide that the procedure is uninformative about the question
at hand. They trace the source of the difference to the failure to include a
sufﬁcient number of lags inthe estimatedVARspeciﬁcations soas toadequately
capture the behavior of hours implied by the underlying theoretical model.
This misspeciﬁcation, in their view, leads to an erroneous conclusion of the
negative response of hours worked to a productivity shock, even though the
data underlying this test are drawn froma standard RBCmodel which implies
the opposite response!
Despite the controversies surrounding the SVARapproach, Basu et al. [32]
present evidence that support the SVAR ﬁndings by generating a modiﬁed
Solow residual that accounts for imperfect competition, non-constant returns
to scale, variable factor utilization, and sectoral re-allocation and aggregation
effects. Unlike the SVAR approach, the evidence obtained from this approach
is robust to long-run identifying assumptions or to the inclusion of new
variables in the estimated dynamic system. They ﬁnd that purging the standard
Solow residual of these effects eliminates the phenomenon of “procyclical
productivity”. They also examine the response of a key set of variables such as
output, hours worked, utilization, employment, non-residential investment,
durables and residential investment, non-durables and services, and various
prices and interest rates to changes in the puriﬁed Solow residual. They use
both standard regression analysis and simple bivariate VARs for this purpose.
Their ﬁndings corroborate the ﬁndings from the SVAR approach regarding
the negative response of hours to technology improvements in the short run.
They also uncover further evidence for the negative response of non-residential
investment to such shocks. Following Gali [96] and Gali and Rabanal [98],
they advance price rigidity as the major reason for these deviations from the
RBC predictions in the short run. These ﬁndings have, on the one hand,
generated substantial controversy and, on the other, cast further doubt on the
ability of the RBCmodel driven by technology shocks to provide a convincing
explanation of economic ﬂuctuations for the major developed countries.
October 9, 2009 15:12 9in x 6in b808-ch06
Chapter 6
Business Cycles in Emerging
Market Economies
In recent years, the real business cycle (RBC) agenda has been increasingly
applied in emerging market contexts. On the one hand, researchers have begun
generating business cycle facts for such economies (see, for example, Ratfai and
Benczur [175, 176]). Onthe other hand, they have beenexamining the efﬁcacy
of RBC-type models in accounting for these facts. Business cycles in emerging
markets exhibit different characteristics compared to those in developed
economies. The recent literature on the “Sudden Stop” phenomenon has
emphasized the large reversals in current accounts and the incidence of
capital outﬂows that have become identiﬁed with many recent emerging
market economies’ experience (see, for example, Arellano and Mendoza [17]).
Emerging market business cycles also display a different set of stylized facts
relative to developed economies. For instance, consumption varies more than
output, the trade balance is strongly countercyclical, and income and exports
are typically highly volatile. The question then arises whether a small open
economy-type RBC model can account for both developed and emerging
market business cycles.
Earlier RBCmodels for small open economies were developed by Mendoza
[162] and Correia, Neves, and Rebelo [77]. Kydland and Zaragaza [144] also
study the behavior of an emerging market economy, namely, Argentina, but
their analysis is based on the one-sector optimal growth model. They argue
that the large shortfalls in GDP suffered by Argentina in the 1980s can be
explained in a simple growth model framework using the observed measure
of productivity or the Solow residual. More recently, Aguiar and Gopinath
101
October 9, 2009 15:12 9in x 6in b808-ch06
102 Business Cycles: Fact, Fallacy and Fantasy
[1] and Garcia-Cicco, Pancrazi, and Uribe [101] have examined versions of
a small open-economy business cycle model with permanent and transitory
shocks to account for emerging versus developed economy experiences.
6.1. A SMALL OPEN-ECONOMY MODEL OF EMERGING
MARKET BUSINESS CYCLES
Aguiar and Gopinath [1] present a small open-economy model of business
cycles that allows for permanent and transitory changes to productivity. In
their view, emerging market economies’ experience can be distinguished by the
large number of regime shifts, here modeled as changes in trend productivity
growth. By contrast, developed economies typically face stable political and
economic policy regimes so that changes to productivity are transitory.
To describe the model, let output be produced according to the Cobb–
Douglas production technology as
Y
t
= exp (z
t
)K
1−α
t
(
t
L
t
)
α
, 0 < α < 1, (1.1)
where {z
t
} and {
t
} represent two alternative productivity processes. The
shock z
t
represents the transitory component of productivity, and evolves as a
stationary AR(1) process:
z
t
= ρ
z
z
t −1
+
z
t
, |ρ
z
| < 1, (1.2)
where {
z
t
}
∞
t =0
is distributed as i.i.d. with E(
z
t
) = 0 and Var(
z
t
) = σ
2
z
.
The permanent shock to productivity evolves as

t
− b
_
− 1
_
, (1.8)
where r
∗
is the world interest rate, b represents the steady-state level of
debt, and ψ > 0 governs the elasticity of the interest rate to changes in
indebtedness. Furthermore, when choosing how much debt to take on, the
individual country does not internalize the fact that an increase in borrowing
will lead to an increase in the interest rate on those loans.
The model expressed in terms of the transformed or “hatted” variables
is solved by using standard recursive methods. Assuming it exists, the value
function deﬁned in terms of the transformed variables is given by
V (
ˆ
K
t
,
ˆ
B
t
, z
t
, g
t
)
= max
ˆ
C
t
,L
t
,
ˆ
K
t +1
,
ˆ
B
t +1
{u(
ˆ
C
t
, L
t
) + f (β, g
t
)E
t
V (
ˆ
K
t +1
,
ˆ
B
t +1
, z
t +1
, g
t +1
)},
where u(
ˆ
C
t
, L
t
) is deﬁned by (1.5) in the case of GHH preferences and by
(1.6) in the case of Cobb–Douglas preferences. Likewise, f (β, g
t
) is given by
βg
1−σ
t
in the former case and by βg
γ(1−σ)
t
in the latter. The optimization is
subject to the transformed budget constraint
ˆ
C
t
+ g
t
ˆ
K
t +1
=
ˆ
Y
t
+ (1 − δ)
ˆ
K
t
−
φ
2
_
g
t
ˆ
K
t +1
ˆ
K
t
− µ
g
_
2
ˆ
K
t
−
ˆ
B
t
+ q
t
g
t
ˆ
B
t +1
.
Substituting for
ˆ
C
t
using the resource constraint, the ﬁrst-order conditions
with respect to
ˆ
K
t +1
,
ˆ
B
t +1
, and L
t
are
u
c
(
ˆ
C
t
, L
t
)
_
g
t
+ φ
_
g
t
ˆ
K
t +1
ˆ
K
t
− µ
g
_
g
t
_
= f (β, g
t
)E
t
_
∂V
∂
ˆ
K
t +1
_
, (1.9)
u
c
(
ˆ
C
t
, L
t
)g
t
q
t
+ f (β, g
t
)E
t
_
∂V
∂
ˆ
B
t +1
_
= 0, (1.10)
u
L
(
ˆ
C
t
, L
t
) + u
c
(
ˆ
C
t
, L
t
)
∂
ˆ
Y
t
∂L
t
= 0. (1.11)
October 9, 2009 15:12 9in x 6in b808-ch06
Business Cycles in Emerging Market Economies 105
The envelope conditions are given by
∂V (
ˆ
K
t
,
ˆ
B
t
, z
t
, g
t
)
∂
ˆ
K
t
= u
c
(
ˆ
C
t
, L
t
)
_
∂
ˆ
Y
t
∂
ˆ
K
t
+ (1 − δ)
+φ
_
g
t
ˆ
K
t +1
ˆ
K
t
− µ
g
_
g
t
ˆ
K
t +1
ˆ
K
t
−
φ
2
_
g
t
ˆ
K
t +1
ˆ
K
t
− µ
g
_
2
_
_
_
,
∂V (
ˆ
K
t
,
ˆ
B
t
, z
t
, g
t
)
∂
ˆ
B
t
= −u
c
(
ˆ
C
t
, L
t
).
For simplicity, consider only the case with GHH preferences. The
stationary competitive equilibrium satisﬁes the following conditions:
(
ˆ
C
t
− τL
t
)
−σ
_
1 + φ
_
g
t
ˆ
K
t +1
ˆ
K
t
− µ
g
__
=
β
g
σ
t
E
t
_
(
ˆ
C
t +1
− τL
t +1
)
−σ
_
1 − δ + exp (z
t +1
)(1−α)g
α
t +1
_
L
t +1
ˆ
K
t +1
_
α
+φ
_
g
t +1
ˆ
K
t +2
ˆ
K
t +1
− µ
g
_
g
t +1
ˆ
K
t +2
ˆ
K
t +1
−
φ
2
_
g
t +1
ˆ
K
t +2
ˆ
K
t +1
− µ
g
_
2
_
_
_
_
_
,
(1.12)
(
ˆ
C
t
− τL
ν
t
)
−σ
=
β (1 + r)
g
σ
t
E
t
_
(
ˆ
C
t +1
− τL
ν
t +1
)
σ
_
, (1.13)
τνL
ν−1
t
= exp (z
t
)αg
α
t
_
ˆ
K
t
L
t
_
1−α
, (1.14)
subject to the production function (1.1), the laws of motion for the shocks
(1.2)–(1.4), the resource constraint (1.7), and the equation describing the real
interest rate (1.8).
The solution for the model is obtained by implementing a log-linear
approximation to the equilibrium conditions described above. In the recent
applications, a subset of the parameters is set at values determined a priori. The
remainder of the parameters, including the parameters of the shock processes,
are estimated using a generalized method of moments (GMM) approach that
we describe in detail in the following chapter.
October 9, 2009 15:12 9in x 6in b808-ch06
106 Business Cycles: Fact, Fallacy and Fantasy
6.2. DO SHOCKS TO TREND PRODUCTIVITY
EXPLAIN BUSINESS CYCLES IN EMERGING
MARKET ECONOMIES?
Aguiar and Gopinath [1] consider two versions of their model, an emerging
market economy version estimated using quarterly data for Mexico and a
developed country version estimated for Canada between 1980 and 2003.
The moments that they use for estimation include the standard deviations of
log (ﬁltered) income, investment, consumption, net exports-to-GDP, as well as
the correlations of the latter three withoutput. They also consider the meanand
standard deviationof (unﬁltered) income growth as well as the autocorrelations
of (ﬁltered) income and (unﬁltered) income growth. This yields 11 moment
conditions. The parameters to be estimated are given by (µ
g
, σ
z
, ρ
z
, σ
g
, ρ
g
, φ).
Since the number of parameters is less than the number of moment conditions,
there are also overidentifying conditions which can be used to test the model’s
implications. The most noteworthy ﬁnding that emerges from the estimation
is that the ratio of shock standard deviations, σ
g
/σ
z
, is 0.25 or 0.41 for Canada
depending on the speciﬁcation for preferences, and 2.5 or 5.4 for Mexico. This
ratio captures the importance of shocks to trend productivity. By contrast,
the autocorrelations of transitory shocks are roughly similar, as is the capital
adjustment parameter φ.
The authors argue that differences in the magnitude of shocks to trend
productivity can account for some of the salient features of business cycles in
emerging market economies. In particular, they showthat a positive transitory
shock to productivity reduces consumption in anticipation of lower income
in the future. By contrast, a positive shock to trend productivity causes
consumption to respond more than output in the expectation that income
will be higher in the future. This leads to a large trade deﬁcit which tends to
persist for a considerable period of time (16 quarters). Thus, the shock to trend
productivity can generate consumption booms that tend to appear side-by-
side withcurrent account deﬁcits inemerging market economies. Furthermore,
these results can also explain why consumption is more volatile than income
in economies where shocks to productivity growth are more important than
transitory shocks.
These ﬁndings have been challenged by Garcia-Cicco et al. [101], who
argue that the results of Aguiar and Gopinath [1] are due to their use of
October 9, 2009 15:12 9in x 6in b808-ch06
Business Cycles in Emerging Market Economies 107
a short sample to estimate low-frequency movements in productivity. By
contrast, Garcia-Ciccoet al. [101] use annual data for Argentina over the period
1913–2005. As we described earlier, much of the business cycle literature for
developed countries has concentrated on the post-World War II period. Given
the evidence that business cycles have moderated during this period, this choice
of sample period does not seem out of line. Garcia-Cicco et al. [101] argue
that a similar ﬁnding is not true for an emerging market economy such as
Argentina. In particular, they show that output ﬂuctuations in the post-World
War II period are as large as those in the pre-World War II period. They
consider a model that is very similar to the one in Aguiar and Gopinath [1],
and estimate the parameters of the stochastic processes for the permanent and
transitory shocks using 16 moment conditions. In particular, they consider
the variances and ﬁrst- and second-order autocorrelations of output growth,
consumption growth, investment growth, and the trade balance-to-output
ratio; the correlations of output growth with consumption growth, investment
growth, and the trade balance-to-output ratio; and the unconditional mean
of output growth.
First, unlike Aguiar and Gopinath [1], Garcia-Cicco et al. [101] ﬁnd that
the overidentifying restrictions of the model are rejected. Second, in terms
of the parameter estimates, the permanent shock is estimated to be more
volatile and persistent than the transitory shock. The standard deviations of the
shocks and the autoregressive parameter for the permanent shock are estimated
precisely. However, this is not the case for the autoregressive coefﬁcient for the
transitory shock, which is not signiﬁcantly different fromzero. As we described
above, consumption growth will be more volatile than output growth if
permanent shocks are more important than transitory shocks, and less volatile
thanoutput growth if the opposite is true. They determine this empirically, and
ﬁnd that the consumption-smoothing motive in response to a transitory shock
dominates the anticipatory effect of consumption to a permanent productivity
shock, and, in contrast to what is observed in the data, consumption growth
in the estimated model is calculated to be less volatile than output growth.
The authors also ﬁnd that the trade balance-to-output ratio is estimated to
be around four times as volatile as output growth, whereas in the data these
quantities are roughly equal. Whereas the ﬁrst result suggests that the estimated
model does not emphasize the role of permanent shocks sufﬁciently, the second
result suggests that it overemphasizes them. Third, investment growth is found
October 9, 2009 15:12 9in x 6in b808-ch06
108 Business Cycles: Fact, Fallacy and Fantasy
to be insufﬁciently volatile, suggesting in this case that neither source of shocks
is sufﬁciently volatile. The authors show that the estimated model also fails
to replicate the autocorrelations of output growth and the trade balance-to-
output ratio. In particular, the trade balance-to-output ratio displays a near
random walk behavior even though the estimated autocorrelation function in
the data is downward-sloping. The authors also show that the random walk
behavior of the trade balance-to-output ratio remains, regardless of whether
shocks toproductivity are permanent or transitory. If the shocks are permanent,
consumption increases in response to an innovation to output in roughly the
same magnitude as output, as households perceive the income increase to be
permanent. Hence, the trade balance is unaffected and the trade balance-to-
output ratio inherits the behavior of output. By contrast, if the shocks are
transitory, the behavior of the trade balance-to-output ratio again follows a
near random walk because in this case, with a constant world interest rate,
consumption in the model follows a near random walk even though output
and investment become stationary variables.
Garcia-Cicco et al. [101] conclude that a pure RBC model driven by
permanent and/or transitory exogenous shifts in productivity does not provide
an adequate explanation of business cycles in emerging markets. In particular,
they argue that some of their ﬁndings that we described above point to the
importance of shocks that are different from productivity shocks. However,
they also argue that their results are derived under the joint hypothesis of
business cycles driven by productivity shocks and the propagation mechanisms
of the standard RBC approach. Hence, their ﬁndings cannot be used to
disentangle which of these factors is responsible for the failure of the model to
replicate the observations.
October 9, 2009 15:12 9in x 6in b808-ch07
Chapter 7
Matching the Model to the Data
In the previous chapters, we described some criticisms leveled against the
assumptions of the real business cycle (RBC) framework. Amongst the most
prominent of these assumptions is the assumption of perfect price ﬂexibility.
We also discussed variations of the model that could be used to account for
speciﬁc correlations in the data. Yet one of the most important aspects of the
debate regarding the RBC approach lies in the use of calibration as a way
of matching the model to the data. In this chapter, we will study alternative
approaches for matching the implications of a theoretical model with the data,
and also provide an overview of the estimation versus calibration debate.
The recent business cycle literature has witnessed the use of a variety of
techniques regarding the empirics of business cycles. One of the most popular
approaches has been based on the dynamic factor model, which seeks to
describe the joint cyclical behavior of a key set of time series in terms of a low-
dimensional vector of unobservable factors and a set of idiosyncratic shocks.
This model was initially proposed by Sargent and Sims [185] for describing
cyclical phenomena. In her original contribution, Altug [5] estimated an
unobservable index model for a key set of aggregate series based on a modiﬁed
version of the Kydland and Prescott model [141] using maximum likelihood.
Watson[209] extendedthis approachtoderive measures of ﬁt for anunderlying
economic model. An alternative approach was proposed by Christiano and
Eichenbaum [65], who used the generalized method of moments (GMM)
approach (see Hansen [113]) to match a selected set of unconditional ﬁrst and
second moments implied by their model. Their approach may be viewed as an
extension of the standard RBC approach, which assesses the adequacy of the
model based on the behavior of the relative variability and co-movement of
109
October 9, 2009 15:12 9in x 6in b808-ch07
110 Business Cycles: Fact, Fallacy and Fantasy
a small set of moments. Canova [56, 57] discusses alternative approaches
for conducting statistical inference in calibrated models. Finally, Bayesian
estimation of the so-called dynamic stochastic general equilibrium (DSGE)
models provides an alternative to incorporating prior beliefs about various
parameters that formalize some of the practices in the calibration approach.
7.1. DYNAMIC FACTOR ANALYSIS
Dynamic factor analysis seeks to describe the joint cyclical behavior of a key set
of time series in terms of a low-dimensional vector of unobservable factors and
a set of idiosyncratic shocks that are mutually uncorrelated and uncorrelated
with the factors. To describe how to formulate unobservable index models, let
{ ˜ w
t
}
∞
t =0
denote an n-dimensional mean zero, covariance stationary stochastic
process used to describe observations on the (possibly de-trended) values of a
set of variables. A k-factor unobservable index model for ˜ w
t
is given by
˜ w
t
=
∞

s=−∞
˜
H(s)
˜
f
t −s
+ ˜ ν
t
, (1.1)
where {
˜
H(s)}
∞
s=−∞
is a sequence of (n ×k)-dimensional matrices,
˜
f
t
is a k ×1
vector of common factors, and ˜ ν
t
is an n × 1 vector of idiosyncratic shocks
that are mutually uncorrelated and uncorrelated with common factors. More
precisely, we require that
E(
˜
f
t
˜ ν
i,t
) = 0 for i = 1, . . . , n (1.2)
E(˜ ν
i,t
˜ ν
j,t
) = 0 for i = j. (1.3)
Both the common factors and the idiosyncratic factors may be serially
correlated, that is, E(
˜
f
t
˜
f
t −s
) = 0 for t = s and E(˜ ν
i,t
˜ ν
i,t −s
) = 0 for all
i, j, t = s. According to this model, covariation among the elements of ˜ w
t
can
arise because they are functions of the same common factor or because they
are functions of different factors which are themselves correlated at different
leads and lags.
Under these assumptions, the variances and autocovariances of the
observed series { ˜ w
t
} can be decomposed in terms of the variances and
autocovariances of a low-dimensional set of unobserved common factors and
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 111
the idiosyncratic shocks. Let
R
w
(r) = E( ˜ w
t
( ˜ w
t +r
)

+ S
ν
(ω),
where
˜
H(ω) denotes the Fourier transformof
˜
H(s). Hence, the dynamic factor
model provides decomposition at each frequency that is analogous to the
decomposition of variance in the conventional factor model. The dynamic
factor model can be estimated and its restrictions tested across alternative
frequencies using a frequency domain approach to time series analysis. The
unrestricted version of the dynamic factor model does not place restrictions on
the matrices
˜
H(s), which describe howthe common factors affect the behavior
of the elements of ˜ w
t
at all leads and lags. Also, it is not possible to identify
the common factors with different types of shocks to the economy.
The use of the dynamic factor model in business cycle analysis dates back to
the work of Sargent and Sims [185]. As these authors observe, dynamic factor
analysis may be linked to the notion of a “reference cycle” underlying the
methodology of Burns and Mitchell [50] and the empirical business cycle
literature they conducted at the National Bureau of Economic Research.
Another well-known application of this approach is due to Altug [5], who
derives an unobservable index model for a key set of aggregate series by
augmenting the approximate linear decision rules for a modiﬁed version of
the Kydland and Prescott [141] model with i.i.d. error terms. Sargent [184]
also employs the device of augmenting a singular model with additional
idiosyncratic shocks. He discusses a classical measurement error case, as in
Altug [5], as well as a case with orthogonal prediction errors. Altug also
uses this representation to estimate the model using maximum likelihood
(ML) estimation in the frequency domain. The restricted factor model makes
use of the cross-equation restrictions across the linear decision rules implied
by the original model. The common factor is identiﬁed as the innovation
to the technology shock, and the idiosyncratic shocks are interpreted as
i.i.d. measurement errors or idiosyncratic components not captured by the
underlying RBC model. Unlike the unrestricted factor model which can be
estimated frequency by frequency, this model must be estimated jointly across
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 113
all frequencies because the underlying economic model constrains the dynamic
behavior of the different series as well as speciﬁes the nature of the unobserved
factor.
2
Altug [5] initially estimates an unrestricted dynamic factor model for
the level of per capita hours and the differences in per capita values of durable
goods consumption, investment in equipment, investment in structures, and
aggregate output. She ﬁnds that the hypothesis of a single unobservable factor
cannot be rejected at conventional signiﬁcance levels for describing the joint
time series behavior of the variables. However, when the restrictions of the
underlying model are imposed, the model cannot explain the cyclical variation
of the observed variables.
7.1.1. Measures of Fit for Calibrated Models
Watson [209] extended the approach in Altug [5] and Sargent [184] to derive
measures of ﬁt for an underlying economic model. Unlike the approach
adopted by Altug and Sargent, Watson’s analysis does not depend on assuming
that the unobserved factors and the idiosyncratic shocks are uncorrelated.
Instead, his approach involves choosing the correlation properties of the error
process between the actual data and the underlying model such that its variance
is as small as possible. Also, the joint process for the data and the error is
introduced to motivate goodness-of-ﬁt measures, not to describe a statistical
model that can be used to conduct statistical tests.
To describe Watson’s approach, let x
t
denote an n ×1 vector of covariance
stationary random variables. Deﬁne the autocovariance generating function
(ACGF) for x
t
by
A
x
(z) =
∞

r=−∞
E(x
t
x
t −r
)z
r
.
Here, {x
t
}
∞
t =0
denotes the stochastic process generated by some underlying
model, and A
x
summarizes the unconditional second-moment properties of
this model. In the data, the vector of variables y
t
corresponds to the empirical
counterpart of x
t
. The ACGF for y
t
is similarly denoted A
y
(z). The question
at hand is whether the data generated by the model are able to reproduce the
behavior of the observed series. For this purpose, deﬁne the n × 1 vector of
2
For further discussion of maximum likelihood estimation in the frequency domain, see Hansen and
Sargent [114].
October 9, 2009 15:12 9in x 6in b808-ch07
114 Business Cycles: Fact, Fallacy and Fantasy
errors u
t
that are required to reconcile the ACGF implied by the model with
that of the data.
3
To continue, deﬁne u
t
by the relation
u
t
= y
t
− x
t
, (1.5)
which implies that the ACGF for u
t
can be expressed as
A
u
(z) = A
y
(z) + A
x
(z) − A
xy
(z) + A
yx
(z), (1.6)
where A
xy
(z) is the joint ACGF for x
t
and y
t
. To calculate A
u
(z), notice
that A
y
(z) can be determined from the data, and A
x
(z) from the model.
However, since A
xy
(z) is unknown, some additional assumptions are needed
to operationalize this approach. In the standard dynamic factor model, the
assumption regarding A
xy
(z) is that it is zero, A
xy
(z) = 0. This implies
a version of a classical errors-in-variables approach.
4
However, in Watson’s
framework, the error term u
t
is the approximation error in describing the
observed data with the underlying economic model. Hence, the assumption
of pure measurement error which is uncorrelated with the true variable is not
appropriate. Nevertheless, a lower bound may be deduced for the variance of
u
t
without imposing any restrictions on A
xy
(z). This bound is calculated by
choosing A
xy
(z) to minimize the variance of u
t
subject to the constraint that
the implied joint ACGF for x
t
and y
t
is positive deﬁnite.
To illustrate this approach, let us consider two cases.
5
In the ﬁrst case,
x
t
, y
t
, and u
t
are assumed to be serially uncorrelated scalar random variables.
The problem is to choose σ
xy
to minimize the variance of u
t
subject to the
constraint that the covariance matrix of x
t
and y
t
remains positive deﬁnite, or
|σ
xy
| ≤ σ
x
σ
y
:
min
σ
xy
σ
2
u
= σ
2
y
+σ
2
x
− 2σ
xy
s.t. |σ
xy
| ≤ σ
x
σ
y
. (1.7)
3
In a standard goodness-of-ﬁt approach, it is the size of the sampling error that is used to judge whether
a given model ﬁts that data. In this case, let A
y
(z) denote the population autocovariance function and
ˆ
A
y
(z)
denote its estimated counterpart. Then, the difference between A
y
(z) and
ˆ
A
y
(z) is ascribed to sampling
error, and the size of the sampling error can be determined from the data-generating process for y
t
. If, in
addition, A
y
(z) = A
x
(z), then the sampling error in estimating A
y
(z) from actual data can also be used to
determine how different A
y
(z) is from A
x
(z).
4
This is similar to the interpretation provided by Altug [5].
5
We omit the third case that Watson considers because it requires results for the Cramer representation
of stationary time series.
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 115
It is easy to see that the solution for this problem is to set σ
xy
= σ
x
σ
y
. As a
consequence, σ
2
u
= (σ
y
− σ
x
)
2
at the minimum. Furthermore, x
t
and y
t
are
perfectly correlated so that
x
t
= γy
t
, (1.8)
where γ = σ
x
/σ
y
.
Now suppose that x
t
and y
t
are serially uncorrelated random vectors with
covariance matrices
x
and
y
. The covariance matrix of u
t
is given by
u
=

y
+
x
−
xy
+
yx
. In this case, we cannot minimize the variance of u
t
directly. Instead, we consider a transformation that allows us to determine
the size of u
t
. One convenient transformation is the trace of
u
, tr(
u
) =

n
i=1

ij,u
, where
ij,u
denotes the i, j element of . An alternative approach
is to minimize the weighted sum of the variances as tr(W
u
), where W is an
n ×n matrix. The problem in this case is to choose
xy
to minimize tr(W
u
)
subject to the constraint that the covariance matrix of (x

t
, y

t
) is positive semi-
deﬁnite. Watson [209] provides a solution for the case in which
x
has rank
k ≤ n so that the number of variables is typically less than the number of
shocks. Proposition 1 in Watson [209] demonstrates that the unique matrix

xy
, which minimizes the weighted sum of the variances of the elements of u
t
subject to the constraint that this matrix is positive semideﬁnite, is given by

= I
k
), and S is a k × k diagonal matrix. An implication of this result is
that, as in the scalar case, the joint covariance matrix of (x

t
, y

t
) is singular. As
a consequence, x
t
can be represented as
x
t
= y
t
, (1.10)
where = C
x
UVC
−1
y
. For k = 1, this result corresponds to the one in the
scalar case, given the properties of the U and V matrices.
Watson [209] uses this methodology to generate goodness-of-ﬁt measures
for a standard RBCmodel with a stochastic trend in technology. Speciﬁcally, he
considers the model in King, Plosser, and Rebelo [131, 132]. He log-linearizes
October 9, 2009 15:12 9in x 6in b808-ch07
116 Business Cycles: Fact, Fallacy and Fantasy
the ﬁrst-order conditions to obtain the solution for log-differences of output,
consumption, investment, and the number of hours worked. His approach
allows for a decomposition by frequency of the variance in each observed
series, the variance explained by the model, and the error in reconciling the
model with the data.
6
He ﬁnds that the biggest differences between the spectra
for output, consumption, and investment occur at frequencies corresponding
to the business cycle periodicities of 6–32 quarters. The model also implies
that the number of hours worked is stationary whereas there is considerable
power at the low frequencies for this series in the data, suggesting that the
stochastic trend properties of the model do not hold in the data. Watson’s
analysis shows that focusing only on a small subset of moments implied by
the model can be misleading because the RBC model is unable to reproduce
the typical spectral shape of economic time series.
7.1.2. Other Applications
Forni andReichlin[93] use the dynamic factor model to describe business cycle
dynamics for large cross-sections. Based on a law of large numbers argument,
they show that the number of common factors can be determined using the
method of principal components, the economy-wide shocks can be identiﬁed
using structural vector autoregression (SVAR) techniques, and the unobserved
factor model can be estimated by using equation-by-equation ordinary least
squares (OLS). They examine the behavior of four-digit industrial output and
productivity for the US economy for the period 1958–1986 and ﬁnd evidence
in favor of at least two economy-wide shocks, both having a long-run effect on
sectoral output. However, their results also indicate that sector-speciﬁc shocks
are needed to explain the variance of the series.
Giannone, Reichlin, and Sala [103] show how more general classes of
equilibrium business cycle models can be cast in terms of the dynamic factor
representation. They also describe how to derive impulse response functions
for time series models which have reduced rank, that is, models for which the
number of exogenous shocks is less than the number of series.
6
A comparison of the spectra implied by the model and those generated by actual data also ﬁgured in
early versions of Altug [5]; see also Altug [4].
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 117
7.2. GMM ESTIMATION APPROACHES
Other papers have employed nonlinear estimation and inference techniques
to match equilibrium business models with the data. Christiano and
Eichenbaum [65] consider the hours-productivity puzzle and use the
generalized method of moments (GMM) approach (see Hansen [113]) to
match a selected set of unconditional ﬁrst and second moments implied by
their model. Their approach may be viewed as an extension of the standard
RBCapproach, whichassesses the adequacy of the model basedonthe behavior
of the relative variability and co-movement of a small set of time series.
Christiano and Eichenbaum [65] derive a solution for the social planner’s
problemby implementing a quadratic approximation to the original nonlinear
problemaround the deterministic steady states. Since there is a stochastic trend
in this economy arising from the nature of the technology shock process, the
deterministic steady states are derived for the transformed variables. Their
estimation strategy is based on a subset of the ﬁrst and second moments
implied by their model. To describe how their approach is implemented, let

1
denote a vector of parameters determining preferences, technology, and the
exogenous stochastic processes. Some of the parameters included in
1
may
be the depreciation rate of capital, δ, and the share of capital in the neoclassical
production function, θ. More generally,
1
= (δ, θ, γ, ρ, ¯ g, σ
µ
, λ, σ
λ
)

T
denote the estimated value of , the
test statistic for the second-moment restrictions is based on the distribution of
F(
ˆ

T
) under the null hypothesis. Using this distribution, the authors show
that the statistic
J = F(
ˆ

T
)

Var[F(
ˆ

T
)]
−1
F(
ˆ

T
) (2.21)
is asymptotically distributed as a χ
2
random variable with two degrees of
freedom.
Using data onprivate consumption, government expenditures, investment,
aggregate hours, and average productivity, Christiano and Eichenbaum [65]
estimate the parameters of the model using GMM and examine the various
unconditional second moments implied by the model. As we described earlier,
the GMM approach has been used in other recent applications. For example,
Aguiar and Gopinath [1] and Garcia-Cicco, Pancrazi, and Uribe [101]
employ this approach in their analysis of business cycles in emerging
markets.
7.3. THE CALIBRATION VERSUS ESTIMATION DEBATE
The crux of the recent business cycle debate centers on how a given theoretical
model should be matched with the data. Kydland and Prescott [141] initiated
this debate in the modern business cycle literature. Since then, there have been
ongoing discussions on both sides of the issue.
7
In this section, we will deal
with a variety of criticisms aimed directly at the calibration approach and some
suggested alternatives to it.
In a highly suggestive analysis, Eichenbaum [82] asked whether RBC
analysis, in fact, constituted “wisdom or whimsy”. He took issue with the
calibration approach by showing that the model’s implications for the variance
of output relative to its value in the data are heavily dependent on assumed
values for the underlying parameters for the technology process. In his words:
Indeed, once we quantify the uncertainty in model predictions arising from
uncertainty about model parameter values, calibrated or otherwise, our view
of what the data is [sic] telling us is affected in a ﬁrst-order way. Even if
7
See Kydland and Prescott [142, 143] for a further discussion and defense of their approach.
October 9, 2009 15:12 9in x 6in b808-ch07
120 Business Cycles: Fact, Fallacy and Fantasy
we do not perturb the standard theory and even if we implement existing
formulations of that theory on the standard postwar sample period and even
if we use the stationary inducing transformation of the data that has become
standard in RBC studies — even then the strong conclusions which mark
this literature are unwarranted. What the data are actually telling us is that,
while technology shocks almost certainly play some role in generating the
business cycle, there is simply an enormous amount of uncertainty of just
what percent of aggregate ﬂuctuations they actually do account for. The
answer could be 70% as Kydland and Prescott [142] claim, but the data
contain almost no evidence against either the view that the answer is really
5% or that the answer is really 200%.
Eichenbaum [82] also presented evidence to show that the performance
of the standard model as well as modiﬁcations that allow for labor hoarding,
for example, deteriorate considerably when a break in the sample is allowed
for. Such a break accounts for the slowdown in productivity growth in the US
that occurred in the late 1960s.
7.3.1. The Dynamics of Output
Cogley and Nason [70] examine the dynamics of output as a way of
determining the efﬁcacy of the RBC model in describing the data. As part
of their diagnostics, they consider the autocorrelation function for output, its
power spectrum, and impulse response functions in response to shocks. Since
the single technology shock model is singular, they use the two-shock versionof
the RBCmodel proposed by Christiano and Eichenbaum[65], which includes
shocks to productivity and government consumption as well as the indivisible
labor feature of Hansen [112] and Rogerson [179]. They assume that the
technology shocks are difference-stationary, whereas government shocks evolve
as a persistent AR(1) process. To estimate impulse response functions from
the data, the authors use the SVAR technique developed by Blanchard and
Quah [43]. They consider a two-variable VAR with output and hours worked
(or consumption), and identify the technology shock under the assumption
that it has a permanent effect on output. They compare the autocorrelation
function and the impulse response function in the data with those generated by
the model using generalized Q statistics. The autocorrelation function from
the model is generated by simulating the model 1000 times. The striking
results in Cogley and Nason [70] show that the autocorrelation function
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 121
for output implied by the model is nearly ﬂat, as is the power spectrum.
8
By contrast, the autocorrelations of output at lags 1 and 2 are signiﬁcant
and positive, and the spectrum for output displays signiﬁcant power at the
business cycle frequencies of 2.33–7 years per cycle. The test statistic also
shows that the implications of the model are rejected at signiﬁcance levels
of 1% or less. In terms of the impulse response functions, the model has
some success in matching the estimated response functions in the data for
the permanent component of GDP, but it is unable to match the impulse
response function for the transitory component. In particular, the data display
a hump-shaped response to transitory shocks whereas the model implies a
much smaller monotonic decay. Cogley and Nason [70] also argue that the
model displays weak propagation mechanisms which arise fromintertemporal
substitution, capital accumulation, and adjustment costs.
7.3.2. Calibration as Estimation
Canova [56, 57] proposes an alternative approach for providing statistical
inference in calibrated models. His approach involves constructing prior
distributions for the parameters used in calibrated models based on existing
estimates in the literature or other a priori information available to the
researcher. To describe his approach, let
¯
X
t
denote a vector stochastic process
with a known distribution that describes the evolution of a set of observed
variables, say, GDP, investment, and interest rates. Also, let X
t
= f (Z
t
, β)
denote the process for these variables generated by a speciﬁc economic theory
or model as a function of exogenous and predetermined variables Z
t
and the
parameters β. Let G(X
t
|f , β) denote the density of the vector X
t
, conditional
on the function f and the parameters β; let π(β|I , f ) denote the density
for the parameters β, conditional on the information set available to the
researcher I and the function f ; and let H(X
t
, β|f , I ) denote the joint density
for the data and the parameters. Denote the predictive density p(X
t
|I , f ) =
_
H(X
t
, β|I , f )dβ. Deﬁne expectations of the functions of the simulated data
8
These results are, in some sense, complementary to those regarding the role of ﬁltering on the cyclical
properties generated from a standard RBC model. See Cogley and Nason [69].
October 9, 2009 15:12 9in x 6in b808-ch07
122 Business Cycles: Fact, Fallacy and Fantasy
(denoted by µ(X
t
)) under the predictive distribution p(X
t
|I , f ) by
E(µ(X
t
)|f , I ) =
_
µ(X
t
)p(X
t
|I , f )dX
t
=
_ _
H(X
t
, β|I , f )dβdX
t
. (3.22)
This approach allows for a probability statement regarding the behavior of
various moments implied by the model. More precisely, suppose that we are
interestedinevaluating Pr(ν(X
t
) ∈ D), where ν(X
t
) is some moment of interest
and D is a bounded set. Then, deﬁne µ(X
t
) = 1 if ν(X
t
) ∈ D and zero
otherwise. Notice also that the function f is typically unknown and must be
obtained using some approximation method. Canova [56] describes how to
account for such an approximation error when computing the moments of
the simulated data. While the densities H and p are unknown, the model can
be simulated repeatedly for different values of β and Z
t
to compute sample
paths for X
t
. In general, the approach that Canova advocates for conducting
statistical inference in calibrated models is as follows:
• Select a density π(β|I , f ) for the unknown parameters, given the
information I available to the researcher and a density k(Z
t
) for the
exogenous processes.
• Draw vectors β from π(β|I , f ) and z
t
from k(Z
t
).
• For each drawing of β and z
t
, generate {x
t
}
T
t =1
and compute µ(x
t
) using
the model x
t
= f (z
t
, β) or its approximation.
• Repeat the above two steps N times.
• Construct the frequency distribution of µ(x
t
) and other statistics of interest
that can be used to evaluate the performance of the model.
One of the key aspects of Canova’s approach is the selection of a density
π that summarizes information about existing parameter estimates in an
efﬁcient manner. This information may be derived from alternative data sets,
model speciﬁcations, or estimation techniques. For example, one could count
estimates of elements of β obtained from alternative studies and smooth the
resulting histogramto obtain the density π(β|I , f ). If such information is hard
to obtain, then one can use a uniform distribution. The calibration approach
involves putting a point mass on a particular value of β. Simulation exercises
conductedafter a subset of the parameters has beenestimated(using GMM, for
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 123
example) involve putting a point mass onβ
∗
estimatedaccording to a particular
method, say, GMM. One can argue that the approach described above is a
global sensitivity exercise that also allows for an evaluation of the model based
on the probabilities attached to events that the researcher is interested in.
7.3.3. Nonlinearity in Macroeconomic Time Series
Another criticism of the RBC model literature is that it has typically
been concerned with examining the ﬁrst- and second-moment properties of
aggregate economic variables for the purpose of matching a model to the data.
Yet there is a new literature that shows that macroeconomic time series may
exhibit marked nonlinear behavior. Such nonlinearities may take the form of
conditional heteroscedasticity suchas ARCHor GARCHeffects. Alternatively,
there may exist asymmetries in various economic variables.
Neftci [166] was among the ﬁrst to demonstrate that unemployment
ﬂuctuations were asymmetrical along the business cycle. Brock and Sayers [47]
tested real macroeconomic variables displayed by deterministic chaotic
dynamics; and while they could not ﬁnd evidence of chaotic behavior, they
nevertheless showed that postwar employment, unemployment, and industrial
production could be described as nonlinear stochastic processes. Ashley and
Patterson [22] developed a test to test for deviation from linear stochastic
processes, either in the form of nonlinear stochastic dynamics or deterministic
chaos. They found strong evidence of nonlinearity in industrial production,
and argued that any reasonable macroeconomic model should display some
form of nonlinear dynamics (see Potter [172] for a review). In our discussion
in Chapter 3, we discussed the role of alternative factors that could give rise to
endogenous changes in productivity such as labor hoarding, variable capacity
utilization, increasing returns, and time-varying markups. In a novel analysis,
Altug, Ashley, and Patterson [9] examine the implied behavior of output,
the factor inputs, and the underlying productivity shocks using a simple
production function framework. Speciﬁcally, they note that observed measures
of output and factor inputs such as the capital stock and hours worked display
marked nonlinear behavior. Using an array of diagnostic tests, they do not
ﬁnd evidence of nonlinearity in the Solow residuals measured after allowing
for increasing returns to scale, markups, or variable capacity utilization. They
conclude that the nonlinearity must lie in the transmission mechanism.
October 9, 2009 15:12 9in x 6in b808-ch07
124 Business Cycles: Fact, Fallacy and Fantasy
Valderrama [207] examines the statistical behavior of national income and
product account aggregates for the US and a set of OECDcountries including
France, Italy, Japan, Mexico, and the UK; and shows that nonlinearities such as
skewness, kurtosis, and conditional heteroscedasticity are common for many
of these aggregates. He argues that standard general equilibrium models are
able to replicate the ﬁrst- and second-moment properties of such variables,
but they are unable to reproduce nonlinearities in these time series. He
poses this as a “canonical” challenge to the RBC approach. Valderrama [207]
considers a simple Brock–Mirman-type growth model with GHH preferences
(see Greenwood, Hercowitz, and Huffman [105]) and adjustment costs in
investment. A value iteration approach is used as the solution procedure. This
ensures that the nonlinearities in the underlying model are not eliminated
through a linearization procedure. The approach to matching the model to
the data is through the efﬁcient method of moments (EMM) developed
by Gallant and Tauchen [99], which is a two-step procedure. In the ﬁrst
step, a seminonparametric (SNP) model is estimated to characterize the
statistical properties of the data. The statistical models are ﬂexible enough
to allow for increasing time dependence in the mean of the process (captured
through a VAR), for conditional heteroscedasticity (ARCH, GARCH), and for
nonnormal disturbances. In the second step, the economic model is simulated
for a given set of parameters. A comparison is made between the statistical
parameter estimates in the SNP step and the statistical parameters obtained
using the simulated data and the same statistical model. Then, the candidate
parameters of the simulated model are adjusted until the simulations of the
economic model have statistical properties similar to those of the data. The
objective function is distributed as a X
2
statistic, as in the GMM approach.
Valderrama [207] selects three statistical models to describe the properties
of the data. The SNP approach nests standard VARs. However, it also
allows for periods of high volatility followed by low volatility (conditional
heteroscedasticity), asymmetric business cycles (i.e., skewness), and excess
volatility (i.e., excess kurtosis). The SNP approach uses a standard VAR to
model the conditional mean and an ARCH-GARCH structure to model the
conditional variance, and it allows for a non-Gaussian error term. The last
feature is achieved by taking a transformation of the normal density using
Hermite polynomials (see Gallant andTauchen[100]). The ﬁrst model selected
by the SNP is a VAR(3) with conditional heteroscedasticity and a Hermite
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 125
polynomial of degree 4. Two other models are selected for the purpose of
describing the data. The ﬁrst of these is a linear VAR(3) so that the statistical
procedure is similar to the RBC approach of matching the impulse responses
from a standard VAR with those generated from the underlying economic
model. The second model is a VAR(3) with ARCH(1) errors, which allows
for nonlinearity in terms of the ARCH effects but continues to assume that
the errors are Gaussian. The parameters of the underlying economic model
are chosen based on standard calibration exercises and also to match the three
statistical models using a simulated method of moments approach.
Valderrama [207] ﬁnds that the biggest difference among the three
statistical models is in terms of the adjustment cost parameter, φ. When the
statistical model is forced to be a linear VAR(3), this parameter is estimated
to be around 10; whereas in the other two models, its estimates are around 3.
The intuition for this result stems from the fact that a high adjustment cost
parameter helps to smooth the implied behavior of investment and to reduce
conditional volatility or kurtosis in the investment series. By contrast, the other
two statistical models allow for these features and hence do not require such a
high adjustment cost parameter. The linear VAR(3) model also implies a much
lower volatility for the consumption and investment series than the other two
models. Valderrama ﬁnds that the RBC model is not successful at generating
the conditional variance of investment; it can capture the nonlinearity in
investment, but not the nonlinearity in consumption. Combined with the
ﬁndings of Altug et al. [9] who show that the nonlinearities in the observed
series must lie in the propagation mechanism, these results suggest that such
features as ﬁnancial frictions or irreversibility in investment are required to
match the nonlinearities of consumption and investment.
Surprisingly, Valderrama [207] ﬁnds that the irreversibility constraint does
not bind during the solution of the model, implying that the irreversibility
feature is not important for matching the model to the data. This is similar
to some earlier results in the literature. Veracierto [208] and Thomas [205]
compare model economies with irreversible investment (in the case of
Veracierto) or lumpy investment (in the case of Thomas) with the actual
economy and with a baseline model of ﬂexible investment, and arrive
at results that indicate little or no signiﬁcant impact of irreversibility or
lumpiness on aggregate investment dynamics. Veracierto [208] argues that the
reason why irreversibility has an impact on aggregate investment in various
October 9, 2009 15:12 9in x 6in b808-ch07
126 Business Cycles: Fact, Fallacy and Fantasy
multi-sector growth models considered in the literature (see, for example,
Coleman [73] or Ramey and Shapiro [174]) is due to their assumption
of unrealistically large sectoral shocks. Similarly, the reason why aggregate
investment displays lower variability in the presence of irreversibility when
ﬁrms are subject to idiosyncratic shocks, as in Bertola and Caballero [42], is
due to their assumption of a very large variance for these idiosyncratic shocks.
Veracierto [208] argues that when the size of sectoral shocks is consistent
with that observed in the data, the aggregate impact of irreversible investment
disappears ina general equilibriumframework. More generally, bothVeracierto
and Thomas ﬁnd that irreversibility or lumpiness at the plant level does not
affect aggregate investment signiﬁcantly once general equilibrium effects such
as endogenous price adjustments are taken into account. However, this is a
puzzling ﬁnding. As Caballero [53] emphasizes: “An important point to note
is that since only aggregate data were used, these microeconomic nonlinearities
must matter at the aggregate level, for otherwise they would not be identiﬁed.”
Yet, our analysis of the simple RBC model with ﬂexible prices and wages
suggests that it cannot successfully match many features of the data. Given the
importance of nonlinearity in macroeconomic time series documented in a
number of studies, we suggest that the reasonwhy irreversibility may not matter
in a model with ﬂexible prices is that the introduction of imperfect competition
with optimal price-setting behavior on the part of ﬁrms or, alternatively,
changing the way in which the labor market is modeled and introducing
wage contracts in the RBCframework may lead to more pronounced quantity
adjustments. Both Veracierto [208] and Thomas [205] assume that labor is
perfectly mobile across plants. This latter feature may compensate for the
rigidity faced by plants given that capital and labor are substitutable according
to the Cobb–Douglas speciﬁcation of technology, and may be responsible
for softening the impact of the inﬂexibility faced by plants in terms of their
capital adjustments. Moreover, as Valderrama [207] states, irreversibility may
play a more important role when ﬁnancial constraints at the plant level and the
household level are taken into account. At the same time, one may consider the
impact of another type of aggregate shock, namely, monetary shocks. Thus,
for example, ﬁnancial accelerator models of investment ﬁnd that ﬁnancially
constrained ﬁrms’ investment responds three times as strongly to a monetary
expansion than that of ﬁrms which are not constrained (see Bernanke, Gertler,
and Gilchrist [41]). These arguments suggest that nonlinearities are another
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 127
area where the assumptions of the simple RBC model fail to hold, and a
potential direction for improving the model’s ﬁt.
7.3.4. The Debate Reconsidered
King [128] discusses the arguments on different sides of the debate under the
heading “Quantitative Theory and Econometrics”. King claims to be on the
quantitative theory side of the debate, and argues against the estimation and
comparison of a “heavily restricted linear time series model (for example, an
RBC model with some or all of its parameters estimated) to an unrestricted
time series model. For some time, the outcome of this procedure will be
known in advance of the test: the probability that the model is true is zero for
stochastically singular models and nearly zero for all other models of interest.”
Returning to the inception of the estimation versus calibration debate, we note
that Altug’s [4] results did not lead to support for the model. Many have argued
that this was to be expected (see Hartley, Hoover, and Salyer [116], p. 17). Yet,
Watson’s [209] insights were partly due to Altug’s initial analysis based on the
spectra implied by the model and those inthe data.
9
Inthe absence of the initial
frequency-domain estimation based on the factor representation, it is unlikely
that RBC models would be subjected to the types of diagnostic tests proposed
by Watson [209]. The factor representation underlying Altug’s analysis has
also been resuscitated by Giannone et al. [103] in a VAR framework.
One can also examine the claim that a less restrictive approach to
estimation and model evaluation following the approach in Christiano and
Eichenbaum [65], for example, may be preferable. King [128] advocates such
an approach as an alternative to calibration that does not face the problems of
“testing highly restrictive linear models”. Yet, one could argue that the GMM
approach in Christiano and Eichenbaum [65] involves choosing a subset of
moments of an equally restrictive nonlinear model. If the simple model is
counterfactual, in what sense does adding another simple feature to such a
contrived model “resolve” a very speciﬁc empirical puzzle? Adding another
shock may “loosen” the behavior of the model, but in what sense does this
make the model a better interpretation of reality? King also discusses the
shortcomings of this approach, highlighting problems in ﬁnding appropriate
9
See Altug [4].
October 9, 2009 15:12 9in x 6in b808-ch07
128 Business Cycles: Fact, Fallacy and Fantasy
instruments and in the appropriate selection of moment conditions for
model evaluation. Gregory and Smith [108] have also argued that the small-
sample properties of the estimators obtained with the approach advocated by
Christiano and Eichenbaum [65] may be far from reasonable if the calibrated
parameters do not consistently estimate the true values.
These arguments show that there is no clear-cut dichotomy between
these approaches. The estimation and testing of a highly restrictive linear
or linearized model may yield many insights regarding the failure of a model
as much as examining the performance of a model based on a small subset of
moments. These developments show that the contribution of the more formal
econometric techniques need not be dismissed as cursorily as is sometimes
the case. Conversely, one could argue that the approach in Christiano and
Eichenbaum [65] is too restrictive. As Geweke [102] notes, examining a small
set of moments does not necessarily lead to a less restrictive approach because
the moments under consideration typically incorporate all the implications
of the underlying theoretical model. In this sense, examining a broader set of
moments may make more sense because this yields more information about
the underlying behavior of the model. One interpretation of quantitative
theorizing is that it helps researchers understand the workings of highly non-
linear dynamic stochastic models and gain some intuition about different
model features. This interpretation precludes the notion that quantitative
theorizing can take the place of formal econometric methods. It is also worth
noting that many of the implications of the standard RBC model have not
withstood the scrutiny conducted under a variety of approaches. The simple
propagation mechanisms inherent in the model have been deemed incapable
of reproducing business cycle dynamics of key variables such as output, and
the New Keynesian challenge has shown that the model fails in generating the
observed negative response of hours to productivity improvements. Another
criticism of the RBC approach is that it fails to capture nonlinearities in key
macroeconomic time series.
Perhaps initially the RBC theorists were concerned that the estimation
versus calibration debate would discredit the use of well-speciﬁed dynamic
general equilibrium models for the purpose of capturing the quantitative
behavior of economic variables. In fact, far fromthis occurring, the estimation
versus calibration debate has led to a panoply of research that has extended
the use of these models in a variety of directions. The initial criticisms of
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 129
the RBC approach have led to a wide set of extensions of the original RBC
approach. As we discuss in the next chapter, the original RBC approach has
eveninstigateda newgenerationof Keynesianmodels that offer features suchas
credit accelerators, sunspot equilibria, and animal spirits. Another extension
of the original approach has been the development of applications that are
useful for policy analysis, a topic to which we turn next.
7.4. DSGE MODELING
Another offshoot of the original RBC debate is the development of dynamic
stochastic general equilibrium (DSGE) models that can be used for policy
analysis. The recent class of models developed for this purpose has vastly more
features than any simple RBC model, and a variety of techniques have been
developed for the purpose of matching the model to the data. One can ask
whether this class of models overcomes some of the criticisms leveled against
the original RBC approach.
Consider, for example, the model recently proposed by Kapetanios, Pagan,
andScott [123] for policy analysis of a small openeconomy. According tothem:
[This] model is stark in its assumptions. There are no market frictions and no
locational speciﬁcity. For example, there is no banking sector and no speciﬁc
role for money and credit in the monetary transmission mechanism. There
are no market frictions and distortions, no ﬁxed costs or discontinuities. The
model assumes a representative household and a symmetric equilibrium for
ﬁrms. Above all, markets are assumed to clear at all times, as all agents have
complete knowledge of the economy and complete understanding of shocks
when they hit. In sum, the model contains assumptions that are almost
guaranteed to be violated by the data, especially in the short run.
Yet this model has a core set of 26 equations!
As another example, consider the model proposed by Christiano,
Eichenbaum, and Evans [67] for the analysis of the effects of monetary policy
shocks on real and nominal variables. The aim of this model is to reproduce
the inertial behavior of inﬂation and persistence in real quantities. To capture
the ﬁrst feature, the model incorporates wage and price contracts following the
approach inCalvo [54]. The model also incorporates a variety of “real” frictions
such as habit persistence in consumption, adjustment costs in investment, and
variable capital utilization. Finally, ﬁrms are assumedto borrowworking capital
to ﬁnance their wage bill. There is also an interest-rate-setting rule that deﬁnes
October 9, 2009 15:12 9in x 6in b808-ch07
130 Business Cycles: Fact, Fallacy and Fantasy
monetary policy. Christiano et al. [67] pursue a limited information estimation
strategy by estimating a subset of the parameters of the model to match the
impulse responses of eight key macroeconomic variables to a monetary policy
shock with those implied from an identiﬁed VAR using the so-called method
of indirect inference.
10
The authors’ analysis is, in many ways, an exploratory
analysis of the role of nominal and real rigidities in propagating shocks. It is
also very much in the spirit of the original Kydland–Prescott approach [141]
in that the goal is to match a set of observed characteristics — in this case, the
impulse responses to a given monetary shock. Christiano et al. provide some
evidence on the role of price and wage rigidities, but in a highly restrictive
environment. For example, the model assumes a continuum of households
which are heterogeneous in their wage and the hours that they work, but
homogeneous in their consumption and asset holdings, with the latter result
derived from the existence of state-contingent securities for consumption.
Christiano et al. [67] derive much of their evidence regarding the types
of real rigidities on the results of calibration-type exercises of the current
generation of macroeconomic models such as habit persistence or adjustment
costs. However, there are shortcomings in their approach. For example,
estimated models of adjustment costs have been shown to imply implausible
degrees of costs of adjustment. Furthermore, the adjustment cost model has
been criticized on the grounds that it implies a constant cost of adjustment
which does not vary with economic conditions. By contrast, the irreversible
investment model studied by Demers [78] and others has been shown to
generate a time-varying adjustment cost that varies in response to changes in
objective and subjective forms of risk and uncertainty.
11
Christiano et al. [67]
assume that monetary policy shocks are drawn from a given and known
distribution. Yet, one could also question the implications of the model
should there be changes in the distribution of the exogenous processes. In
such cases, the assumption of a constant adjustment cost parameter would
fail to hold, implying that the propagation mechanisms would vary with
changes in the distribution of exogenous variables facing agents. This raises
10
For a recent example of the method of indirect inference applied to a model of the EU economy,
see Meenagh, Minford, and Wickens [160].
11
For a review and discussion, see also Demers, Demers, and Altug [79]. For applications, see Altug,
Demers, and Demers [10, 11].
October 9, 2009 15:12 9in x 6in b808-ch07
Matching the Model to the Data 131
the issue of whether DSGE models are, in fact, structural or invariant to
interventions.
Following the initial contribution of Altug [5], a variety of other papers
have implemented maximum likelihood estimation of dynamic equilibrium
models. McGrattan, Rogerson, and Wright [159] consider an equilibrium
business cycle model with household production and distortionary taxation.
As in the analysis of Altug [5], they augment the approximate linear laws of
motion with additional shocks. However, they employ time domain methods
by making use of the Kalman ﬁlter with a linear law of motion for the state
variables and a linear measurement equation for a key set of observables.
More recent applications include Canova [60] and Ireland [121], amongst
others. The recent research on DSGE models has also employed Bayesian
analysis as a way of incorporating prior uncertainty about the parameters of
interest (see, for example, Schorfhiede [186] and An and Schorfhiede [15]).
Similar to Canova [56], this approach ﬁrst provides a link with the calibration
approach by allowing the researcher to use information from microeconomic
studies or macroeconometric estimates. Second, the use of Bayesian methods
and, in particular, examining the posterior distribution calculated as the
prior distribution times the likelihood function is often more straightforward
computationally than maximizing the likelihood function, which is typically
a high-dimensional object.
The Bayesian estimation of DSGE models involves several steps. First, the
method presupposes that a solution can be found for the underlying economic
model of interest. There are a variety of approaches for solving nonlinear
dynamic stochastic models. Judd [122] provides a textbook treatment of
many currently used solution methods. Second, the likelihood function for
the observations must be formed. Third, the posterior distribution for the
parameters, which is obtained from the prior distribution and the likelihood
function, must be examined. To illustrate this approach, suppose that the state
space representation for the model’s solution consists of the following:
• a transition equation S
t
= g(S
t −1
, v
t
, θ), where S
t
is a vector of state
variables that describe the evolution of the model over time, v
t
is a vector
of innovations, and θ is a vector of parameters characterizing preferences,
the production technology, and information; and
October 9, 2009 15:12 9in x 6in b808-ch07
132 Business Cycles: Fact, Fallacy and Fantasy
• a measurement equation Y
t
= h(S
t
, w
t
, θ), where Y
t
are the observables
and w
t
are the shocks to the observables (which may take the form of
measurement errors, among others).
This state space representation can be used to derive the likelihood function
of the observables and to obtain the posterior distribution for the parameters
conditional on the observations. Given the probability density functions for
the shocks f
v
( · ) and f
w
( · ), we can also compute the probabilities p(S
t
|S
t −1
; θ)
and p(Y
t
|S
t
; θ). Notice that
Y
t
= h(g(S
t −1
, v
t
, θ), w
t
, θ).
Hence, we can compute p(Y
t
|S
t −1
; θ). Likewise, the state space representation
allows us to compute the likelihood of the observations y
T
= (y
1
, . . . , y
T
) at
the parameter values θ as
p(y
T
; θ) = p(y
1
|θ)
T

This debate continues to this day.
v
. in particular — is affected by major changes in economic conditions. In 2004. and The Royal Swedish Academy of Sciences published a report titled Finn Kydland and Edward Prescott’s Contribution to Dynamic Macroeconomics: The Time Consistency of Economic Policy and the Driving Forces Behind Business Cycles [204].Preface
How do intellectual disciplines progress? Undoubtedly. This book draws upon Kydland and Prescott’s original contribution. I was a Ph. The Great Depression greatly inﬂuenced the perceptions of a generation of economists. Sometimes. Finn Kydland and Edward Prescott received the Nobel Prize in Economics. real business cycle (RBC) analysis has come to provide a ﬂexible and popular approach for examining macroeconomic phenomena. In recent years. The model was rejected. beginning with Keynes. Large-scale computers in the post-World War II era played an important role in the development of simultaneous equation models. student at the Graduate School of Industrial Administration at Carnegie Mellon University when Kydland and Prescott’s “Time-to-Build and Aggregate Fluctuations” article was published in the early 1980s [141]. the discipline of economics — and macroeconomics.D. The ﬁeld of macroeconomics has changed signiﬁcantly due to Kydland and Prescott’s contributions. much to the delight of macroeconomists of a more Keynesian bent! Yet many felt that economic models should be subject to formal econometric and statistical testing. the development of new techniques or new ways of modeling can also affect the course that a discipline takes. The oil shocks of the 1970s and the 1980s affected economists’ views regarding the sources of macroeconomic ﬂuctuations. My thesis was on estimating the model in the same article.

and Kevin Salyer’s [116] collection of articles provides a critique of the calibration approach. Not content with the initial rejection of the model. Jordi Gali’s [97] recent text articulates an alternative New Keynesian framework for describing aggregate ﬂuctuations. There have been a number of excellent publications that have examined different facets of this literature. Fallacy and Fantasy
Kydland and Prescott’s seminal article initiated the school of RBC analysis. The volume by Thomas Cooley [74] can be considered a primer of RBC analysis and its applications. reﬂects my own interests. many researchers also pursued the econometric analysis of RBC models.vi
Business Cycles: Fact. researchers at central banks have begun using so-called dynamic stochastic general equilibrium (DSGE) models for policy analysis. This book attempts to provide an overview of the burgeoning business cycle literature that. asking some basic questions about RBC analysis and summarizing the ongoing controversies surrounding it. This literature evolved in different ways. in many ways.
. In recent years. This book takes a more eclectic approach. James Hartley. Talented and creative individuals extended the initial Kydland–Prescott research in different ways. Kevin Hoover.

However. He identiﬁed three long waves: 1780–1840. there were periodic movements or “long waves” in economic variables. other scholars investigated methods for the systematic
1
. 1840–1890.” Ever since. and crises experienced by market economies in the 19th and early 20th centuries. and 1890–1950. both growth and business cycles could be ascribed to the process of innovation. Historically. As Mitchell [163] recounts. chemical processing. This book describes these new developments. Kondratiev [137] argued that in addition to shorter economic cycles.Chapter 1
Introduction
In the opening page of the book Business Cycles published in 1927. corresponding to the Industrial Revolution. While some scholars proposed different theories to explain ﬂuctuations in economic activity. In his framework. Schumpeter [187. more effort is required to master it. there have been many developments in the ﬁeld of business cycles. much effort was devoted to understanding the causes of what many viewed as “abnormal” phenomena. 188] sought to explain the existence of such long waves as an outcome of technological innovations. corresponding to the introduction of steel and steam engines. when one examined the history of commercial cycles for the capitalist economies of the time. In the 1920s. corresponding to the invention of electricity. and motor engines. one of the most prominent thinkers of the time. Wesley Mitchell [163] comments as thus: “As knowledge of business cycles grows. According to Karl Marx. “crises” are an endemic feature of capitalist economies. the notion of business cycles originated from various types of panics. depressions. other economists observed that the alternating phases of prosperity and depression seemed to follow each other on a regular basis.

and modeled business cycles as the response of a second-order dynamic system to random shocks. led researchers to account for the observations using new mechanisms for the effects of money on output. Burns and Mitchell [50] and researchers at the National Bureau of Economic Research (NBER) began to identify the phenomena of business cycles. The Great Depression and World War II were two major events in the development of business cycle analysis. Writers such as Frisch and Slutsky embedded the notion of business cycles into simple dynamic systems driven by stochastic shocks. Their ideas were formulated in terms of linear time series models. Keynes’ General Theory [127] laid the foundations for the analysis of short-run economic ﬂuctuations. Post World War II. the Keynesian framework was interpreted as a model of output determination at a point in time. post World War II. During this time.1 We discuss the NBER methodology and the stylized facts of business cycles in Chapter 2.
. According to them. the lessons of the Great Depression were vivid in the minds of many policy-makers. 149] developed monetary models of the business cycle as a way
1 See also Zarnowitz [211]. Fallacy and Fantasy
measurement and identiﬁcation of business cycles. Nevertheless. Slutsky [193] argued that the sum of a number of uncorrelated shocks could produce serially correlated or smooth movements in the generated series.2
Business Cycles: Fact. Their inﬂuence on thinking about business cycles has persisted to this day. The Norwegian economist Ragnar Frisch [95] developed the notions of impulse and propagation mechanisms for describing business cycles. Another important channel which affected the study of business cycles was the development of statistical and time series methods. which continue to form the main vehicle for empirically studying business cycles. Phelps [170] and Lucas [148. After the early work of Burns and Mitchell. In their seminal contributions. the focus shifted to stabilization policy. Their work involved the dating of business cycles and the development of leading indicators for the US economy. The oil shocks of the 1970s and the experience of high inﬂation and high unemployment. and it continues to this day in the business cycle dating methodology employed by the NBER. a business cycle is the simultaneous downturns and upturns of a large number of economic series. The period following World War II was an era of high and sustained growth in many countries. or stagﬂation as it is popularly known.

time-tobuild in investment. During this period. the merits of this approach have been a topic of debate. Since this book purports to discuss business cycles. One of the major issues with Kydland and Prescott’s contribution. We will discuss RBC models further in Chapter 3. the debate on the empirical validation of business cycle models. the emphasis was on generating the Phillips-curve type of phenomena between inﬂation and unemployment based on informational frictions. In some sense. Widely known as the calibration approach. Lucas and Rapping [153] argued that observed ﬂuctuations in aggregate labor supply could be modeled as the voluntary response of agents based on intertemporal substitution effects. we cannot ignore the calibration approach or. The literature on real business cycle (RBC) theory owes its existence to Kydland and Prescott [141]. Lucas [151] cataloged the remarkable conformity in a set of economic series and set forth an agenda for explaining these facts using an equilibrium approach. more generally. unanticipated shocks to money. this feature of Kydland and Prescott’s analysis
. namely. Despite providing great theoretical advances in the analysis of aggregative phenomena. this involves matching a small set of moments implied by the model with those in the data. Long and Plosser [147] developed a simple Robinson Crusoe economy and generated many of the characteristics of modern macroeconomic time series through the mechanisms of substitution and wealth effects in response to technology shocks affecting different sectors of the economy. Though this approach is cited extensively. namely. and inventories. the speciﬁc mechanisms postulated in this literature as leading to business cycles. there was a revival of interest more generally in examining aggregate economic activity as recurrent phenomena characterizing the functioning of economies with optimizing agents. who presented a model that featured technology shocks as the main impulse behind cyclical ﬂuctuations and proposed a rich array of propagation mechanisms for these shocks.Introduction
3
of providing a consistent theoretical foundation for describing the impact of changes in money on output. as identiﬁed by skeptics. the durability of leisure. In Phelps’ and Lucas’ framework. was identifying technology shocks that could generate cyclical ﬂuctuations of magnitudes observed in the data (see Summers [202]). failed to garner sufﬁcient empirical support. In his article “Understanding Business Cycles”. They also proposed a methodology for confronting their theory with the data.

For example. Productivity. is known to be procyclical. The New Keynesian viewpoint breaks with the RBC approach by contesting the view that prices adjust frictionlessly to clear markets. Other skeptics contested the implications of the RBC approach for the observed behavior of productivity. and international capital ﬂows. models have been developed that study the role of market completeness/incompleteness on cyclical ﬂuctuations (see. In a series of papers. the observed procyclical movements in productivity should merely be a response to exogenous technology shocks (see Prescott [173]). international risk sharing. We will examine a few of these directions in later chapters. It also failed to account for the correlation between hours and productivity. for example. or the Solow residual. Hall [110. it introduces alternative mechanisms for generating price stickiness such as imperfect competition among ﬁrms. hence. The RBC literature has also generated international business cycle models to replicate ﬁndings on current account dynamics. Many of these issues have taken on the character of “puzzles” in the RBC literature. The original RBC framework has also been extended in recent years to examine the business cycle phenomena in emerging market economies. ﬁnancial diversiﬁcation. 111] argued persuasively that there were most likely endogenous components to the cyclical movement of productivity arising from imperfect competition at the ﬁrm level and internal increasing returns to scale in production. the original Kydland–Prescott model could not explain the relative variability of hours and productivity or real wages. endogenous changes in efﬁciency due to increasing returns to scale. Instead. The model lacked money. A deeper problem lay in modeling the movement of economy-wide averages (possibly fallaciously) in terms of the behavior of a representative or stand-in household. markups. it faced the problem of reconciling observations on money–output correlations within a model where the main driving force was real productivity shocks. and variable factor
. On the one hand. A more general critique to RBC analysis was mounted by the New Keynesian challenge.4
Business Cycles: Fact. According to the RBC approach. a plethora of papers have presented modiﬁcations to the original Kydland– Prescott framework to reconcile the model with many of the actual features of the data. and they have constituted the topic for much further study. More recently. Fallacy and Fantasy
was viewed as “fantastical” by many. Heathcote and Perri [117]). Subsequent research that evolved from the original Kydland–Prescott exercise has proceeded along several different dimensions.

2 We will discuss the issues involved in matching the model with the data in Chapter 7. This facilitates the analysis of the different effects of government versus technology shocks.
2 Canova [58] is an excellent reference source on the quantitative and empirical analysis of dynamic stochastic general equilibrium models. and Kimball [32]). for example. Following Sims [191]. More recently. Smets and Wouters [194. or monetary versus technology shocks. Fernald.Introduction
5
utilization. vector autoregression (VAR) and structural VAR (SVAR) have also proven to be popular in empirical macroeconomic research. The controversy over the calibration approach also resulted in work on alternative methods for empirically analyzing business cycle phenomena. While both models allow for rich dynamic interrelationships among a set of endogenous variables and an examination of business cycle dynamics based on impulse response functions. 195]). The New Keynesian challenge has proceeded along theoretical lines (see Rotemberg and Woodford [182]) and empirical considerations (see Gali [96] or Basu. SVAR also permits an identiﬁcation of shocks. One approach that is popular in the business cycle literature is the method of unobservable index models or dynamic factor analysis developed by Sargent and Sims [185] and others. dynamic stochastic general equilibrium (DSGE) models have been developed to identify shocks and propagation mechanisms of business cycles models (see.
. We will discuss New Keynesian models in detail in Chapter 5.

This page intentionally left blank
.

Chapter 2
Facts
The vast literature on business cycles has focused on generating the stylized facts regarding cyclical ﬂuctuations. This framework is based on the principle of identifying turning points in economic activity and determining which series constitute leading. we describe the National Bureau of Economic Research (NBER) methodology for dating business cycles and present the basic facts regarding business cycles. and Stock and Watson [199]. and Burns and Mitchell [50] provide a framework to describe the main features of business cycles. We will also discuss some of the empirical ﬁndings in these regards. or lagging indicators of the business cycle. and Osborn [20]. Basu and Taylor [31] have examined business cycles in an international historical context.
2.1. among others). Stock and Watson [197] present a modern methodology to describe business cycles in terms of the cyclical time series behavior of the main macroeconomic series and their comovement with cyclical output. Mitchell and Burns [164]. Mitchell [163]. the European or euro area business cycle has also been the topic of much recent study (see Artis. Kontolemis. coincident. In this chapter. Much of the early work on business cycles was implemented for the US economy. DEFINING A BUSINESS CYCLE
The notion that market economies are subject to recurring ﬂuctuations in a large set of variables was formalized by Burns and Mitchell [50] in their 1946 work entitled Measuring Business Cycles as follows:
Business cycles are a type of ﬂuctuation found in the aggregate economic activity of nations that organize their work mainly in business enterprises. Artis and Zhang [18]. a cycle consists of expansions occurring at about the same time in many
7
. However.

8
Business Cycles: Fact. how broadly should the aggregates that are being considered be deﬁned? The notion that changes in economic activity occur “at about the same time” admit the possibility of economic variables that lead or lag the cycle. and amplitude of each speciﬁc cycle are compared with that of the reference cycle. this sequence of changes is recurrent but not periodic. and second. As part of this analysis. Some of these questions are precisely the ones that we seek to answer in this book. they are not divisible into shorter cycles of similar cycles with amplitudes approximating their own. and revivals which merge into the expansion phase of the next cycle. 14):
. the comments regarding the duration and amplitude of business cycles are based on actual observations of cyclical phenomena. followed by similarly general recessions. If the answer to the latter question is in the afﬁrmative. p. in duration business cycles vary from one year to ten or twelve years. determine whether these turning points are sufﬁciently common across the series.
This deﬁnition has formed the basis of modern thinking about business cycles. Fallacy and Fantasy
economic activities. Mitchell and Burns [164]. when one considers the statement regarding expansions occurring in “many economic activities”. then an aggregate business cycle or a reference cycle is identiﬁed. then should one worry about differences in business cycle activity across regions? Should business cycles be considered in an international context? How about the historical nature of business cycles? Have business cycles moderated over time? Likewise. random ﬂuctuations. 2. whether it pertains to the measurement of business cycles or the construction of models of cyclical ﬂuctuations. contractions. ﬁnd the cyclical peaks and troughs in a given set of economic variables. In seeking to identify “recurrent changes”. The NBER approach to identifying business cycles as outlined by Mitchell [163]. Burns and Mitchell [50] themselves noted that this deﬁnition raised as many questions as it sought to answer. the duration. Burns and Mitchell [50] stress that the notion of a reference cycle should not be equated with an observable construct. In their words (Burns and Mitchell [50]. timing. and also lay down rules for excluding irregular movements and other similar changes. Ch. If one talks about “ﬂuctuations in the aggregate economic activity of nations”. how should we deal with seasonal changes. the cyclical behavior of each series is then examined relative to the reference cycle. or secular trends? Finally. and Burns and Mitchell [50] is comprised of two mutually reinforcing acts: ﬁrst. Once the reference dates are found.

There is an alternative approach which considers the decline in the series measured as a deviation from its long-run trend. classical cycles typically have recessions that are shorter than expansions because of the growth effect.1 displays the difference between classical and growth cycles. This is known as a classical business cycle. like other concepts. What we literally observe is not a congeries of economic activities rising and falling in unison. a trough occurs at point C for a growth cycle while point D deﬁnes a peak.
Classical and Growth Cycles. Point A deﬁnes a trough for a classical cycle while point B deﬁnes a peak. we say that it is
Fig.Facts
9
When we speak of ‘observing’ business cycles we use ﬁgurative language.1. but changes in readings taken from many recording instruments of varying reliability.
The NBER business cycle methodology identiﬁes a business cycle based on the (absolute) downturn of the level of output. By contrast. Following the terminology in Zarnowitz [211]. When the economy is moving from a trough to a peak. Figure 2.
. such cycles are known as growth cycles. By contrast. 2. One advantage of using growth cycles is that they have expansions and contractions that are approximately of the same duration. For. business cycles can be seen only ‘in the mind’s eye’.

As King. Post WWII. The amplitude of a business cycle is the deviation from trend. in their survey of empirical business cycles. Stock. The contribution of Nelson and Plosser [167] was to show that economic time series such as real GDP typically possess unit roots. the average length of an expansion has increased to 57 months from. the average length of a contraction has decreased to ten months from 17 months or more in the years preceding 1945. Plosser.10
Business Cycles: Fact. There are 32 complete cycles in the entire sample period. 38 months pre-WWII. Similarly. The dating of business cycles for the US is done formally by the NBER Business Cycle Dating Committee. We can also observe expansionary and contractionary phases of the business cycle from this table. As an example. this committee recently announced that the US economy had formally been in a recession since December 2007. employment. There are several approaches to de-trending economic time series. If one chooses to use growth cycles. and trade at the sectoral and aggregate levels to identify and date business cycles. The average length of a cycle (peak from previous peak or trough from previous trough) in the post-WWII era is 67 months or over ﬁve years. One of the longest expansions to have occurred post-WWII is between March 1991 and November 2001 — a period of 120 months. The shortest cycle is 17 months or nearly six quarters. This committee uses data on real output. equivalently. national income.1 gives the dates of business cycles or the so-called “reference dates” for the US economy since 1857. Stock and
. quarters. real business cycle (RBC) models which allow for trends in the technology shock imply that growth and business cycles are jointly determined. two peaks. This period was dubbed as the period of the “Great Moderation”. the practice of separating the trend and cyclical component using linear time series methods is well established. The turning points are determined judgmentally. and a recession is said to occur when the economy is moving from a peak to a trough. or years) that the economy spends between two troughs or. Nevertheless. The duration of the business cycle is the length of time (in months. there is an issue of how to identify the cyclical component of a given series. One approach is to use a linear de-trending procedure which assumes that the underlying series possesses deterministic time trends. Table 2. Fallacy and Fantasy
in an expansion. However. at most. and the longest cycle is 128 months or more than ten years. An alternative approach is to assume a stochastic trend modeled as a unit root in the series at hand. and Watson [133] argue. although a computer algorithm exists that can approximate the results (see Bry and Boschan [49]).

whereas the second exacerbates the role of short-term noise. the ﬁlter is applied using a ﬁnite sample.
2
The cyclical component of the series is deﬁned as ytc = yt − gt . if the underlying data are
.2)
In practice.1). let yt denote the series in question and gt denote the unknown trend component. including Singleton [192]. Speciﬁcally. (1. which involves minimizing a quadratic form to determine the trend component in a given series. The Hodrick–Prescott ﬁlter deﬁnes gt to solve
∞ ∞
min
gt t=−∞
(yt − gt ) + µ
t=−∞
2
(gt+1 − gt ) − (gt − gt−1 ) . is modiﬁed at the beginning and end of the sample. then the de-trending procedure has favorable properties. µ is typically recorded as 1600. (1.Facts
13
Watson [197] argue that neither linear time trends nor ﬁrst-differencing to eliminate unit roots provides a satisfactory approach to identifying the cyclical component of a series. One of these approaches is the so-called Hodrick–Prescott [120] (HP) ﬁlter. If these are trend-stationary. Eq. 1 + µ(1 − L−1 )2 (1 − L)2 (1. King and Rebelo [130]. Taking the derivative with respect to gt yields
∞ ∞
µgt+2 − 4µgt+1 + (1 + 6µ)gt − 4µgt−1 + µgt−2 =
t=−∞ t=−∞
yt . Several alternative approaches have been proposed in the recent business cycle literature to separate the trend versus cyclical component of a time series.1)
Using the lag operator notation and simplifying. Cogley and Nason have argued. For quarterly data. the cyclical component can be represented as ytc = yt − gt = µ(1 − L−1 )2 (1 − L)2 yt . and Cogley and Nason [69]. The properties of the HP ﬁlter have been studied by many authors. The general equation characterizing the ﬁlter. in particular. that business cycle dynamics obtained by using a HP de-trending procedure depend on the properties of the underlying data. The parameter µ controls the smoothness of the trend component. The ﬁrst approach tends to generate spurious business cycle effects in the de-trended series.

e. namely. Baxter and King derive the band-pass ﬁlter by considering low-pass and high-pass ﬁlters with the required properties. however. (1. i.5s and 8s. . A second approach is based on the spectral analysis of economic time series. . Let s denote the number of observations in a year.5s)). However. periodicities between six quarters and eight years. once the ideal ﬁlter’s weights are found in the time domain. The frequency response function of the ideal band-pass ﬁlter is deﬁned as βbp (ω) = I (2π/(8s) ≤ ω ≤ 2π/(0.14
Business Cycles: Fact. application of the HP ﬁlter induces spurious business cycle ﬂuctuations. In particular.4)
where ψ(L) is a (K − 1)-order symmetric moving average polynomial. I (2) or less.
(1. . then it has trend elimination properties. Hence. . where I (·) is the indicator function. ak = a−k for k = 1. k=K they show that the lag polynomial describing the K th-order moving average can be written as a(L) = (1 − L)(1 − L−1 )ψ(L). Fallacy and Fantasy
difference-stationary. the optimal
. The band-pass ﬁlter of Baxter and King [37] is obtained by applying a K th-order moving average to a given time series: yt∗
K
=
k=K
ak yt−k . K . The ﬁlter is designed to have a number of other properties. K ak = 0. including the property that the results should not depend on the sample size and that it does not alter the timing relations between series at any frequency. The band-pass ﬁlter retains the movements in a series for the frequencies associated with the periodicities of 0. the Baxter–King ﬁlter will eliminate deterministic quadratic trends or render stationary series that are integrated up to order two.3)
where the moving average coefﬁcients are chosen to be symmetric. It turns out that the time-domain representation of the band-pass ﬁlter is an inﬁnite moving average. The band-pass ﬁlter developed by Baxter and King [37] “ﬁlters” out both the long-run trend and the high-frequency movements in a given time series while retaining those components associated with periodicities of typical business cycle durations. They show that if the sum of the moving average coefﬁcients is zero.

2 illustrates the logarithm of real GDP for the US and its trend and components estimated according to the Baxter–King ﬁlter for the period 1947(I)–2005(III).0 7. showing the large gains in productivity and growth attained over this period. we use this approach to generate the cyclical components of the different series. We consider a measure of the business cycle as the co-movement of the cyclical behavior of individual series with
. Figure 2.00 0. We observe that real GDP displays a marked positive trend over the sample period.4 -0.6 9.01 8.02 -0. 2.03
15
50 55 60 65 70 75 80 85 90 95 00 05 BPCYCLEUSA
Fig.05 -0.2. Trend and Cyclical Components in USA GDP. we provide some stylized facts of business cycles.2 discusses these properties of business cycles. The cyclical component tracks very well the US business cycle as identiﬁed by the NBER. Baxter and King [37] employ a ﬁnite-order approximation to obtain the coefﬁcients of this ﬁlter with a truncation of K = 3 years or K = 12 quarters. Nevertheless. We also observe the cyclical troughs and peaks of economic activity associated with the business cycle. These facts constitute important benchmarks by which to judge the performance of alternative models of business cycles. STYLIZED FACTS
In this section. We can observe the cyclical downturns and upturns corresponding to the NBER business cycle reference dates from this graph.02 8.
2.04 0.8 8.01 0.
approximating ﬁlter with maximum lag K is obtained by truncating the ideal ﬁlter’s weights at lag K . In what follows.2.2 0.2 50 55 60 65 70 75 80 85 90 95 00 05 BPTEST USA -0.Facts
9.04 7.6 -0. Section 2.03 0. the notion of a “business cycle” is also concerned with the co-movement of a large number of economic variables.

In other words. a leading indicator reaches a peak before real GDP reaches its peak and bottoms out (reaches a trough) before real GDP. 2.
. Finally. According to this approach. They use the HP ﬁlter to derive the cyclical components of the different series.1 Information on leading indicators is also collected by the Conference Board. inventories. Fallacy and Fantasy
the cyclical component of real output. Backus and Kehoe [23] analyze the properties of historical business cycles for ten developed countries using a century-long dataset up to the 1980s. Leading indicators are useful for predicting subsequent changes in real GDP. and separate their sample period into the pre-World War I era. the interwar era between World War I and II. Exceptions are production of agricultural goods and natural resources. lagging indicators reach a peak or trough after real GDP. and passed over to the Conference Board in 1995. The stylized facts of business cycles are typically deﬁned in terms of the behavior of the main components of GDP. and monetary aggregates.16
Business Cycles: Fact. which are not especially procyclical. investment. asset returns and prices. They make use of the Baxter–King band-pass ﬁlter to identify the trend versus cyclical components of each series. Variables that move in the opposite direction (rise during recessions and fall in expansions) are countercyclical. and the cross-correlations of the cyclical component of GDP with the cyclical component of each series for lags and leads up to six periods. For example. Among other measures. the contemporaneous correlation of output in different sectors of the economy is large and positive. Coincident indicators reach a peak or a trough at roughly the same time as real GDP. and the post-World War II era. We can also examine whether different time series are out of phase with real GDP. Consumption of durables is much more volatile than consumption of nondurable goods and services. Such a composite index was made ofﬁcial by the US Department of Commerce in 1968.2 The salient facts of a business cycle can be described as follows: 1. Consumption. Consumption of
1 Stock and Watson [197] use data on 71 variables to characterize US business cycle phenomena over the period 1953–1996. Real output across virtually all sectors of the economy moves together. productivity. 2 Early work on developing a composite index of leading indicators is due to Mitchell and Burns [164]. Variables that display little correlation with output over the cycle are called acyclical. and imports are all strongly procyclical. These have been described in a number of economic studies. they consider the co-movement of the cyclical component of GDP with the cyclical component of each series. real wages. variables that move in the same direction over the cycle as real output are procyclical. hours.

Real wages are procyclical or acyclical. The implication is that most ﬂuctuations in total hours result from movements in and out of the work force rather than adjustments in average hours of work. Proﬁts are highly volatile. 10. an inverted yield curve. 13. Investment in equipment and nonresidential structures is procyclical with a lag. They have not displayed a steady pattern in terms of variability to GDP or in terms of leading.
9. 4. The correlation between government expenditures and output is nearly zero. Productivity is slightly procyclical. The employment series lags the business cycle by a quarter.
6. Employment ﬂuctuates almost as much as output and total hours of work.Facts
17
3. Nominal interest rates tend to be procyclical. Total employment. The risk premium for holding private debt. Net exports are countercyclical. Investment in residential structures is procyclical and highly volatile. Since imports are more strongly procyclical than exports.
7. That is. whereas nondurables ﬂuctuate considerably less. or the yield spread between corporate paper and Treasury bills with six months’ maturity. namely. The reason for this countercyclical behavior is likely to be changes in default risk. 5. an expansion is characterized by expectations of higher interest rates at longer horizons whereas a recession typically signals a decline in long-term interest rates relative to short-term rates. but both real wages and productivity vary considerably less than output. The correlation with output is generally negative. Velocity and the money supply are procyclical. while capacity utilization tends to be coincident. 11. tends to shrink during expansions and increase during recessions. while average weekly hours ﬂuctuate much less. or lagging indicators. employee hours. The yield curve which shows the rates of return on bonds of different maturities tends to be upwardsloping during an expansion and downward-sloping at the onset of a recession. coincident.
durable goods ﬂuctuates more than GDP. the trade balance tends to be countercyclical.
8.
. Government spending tends to be acyclical. and capacity utilization are all strongly procyclical. but weakly so.
12.

with the exception of Germany and Japan. there is more evidence of supply-side-driven ﬂuctuations in output. Other papers have generated business cycle facts using data over long samples. The correlation is typically larger in the post-war period than in the pre-war period. In the postWWII era. Inﬂation is a coincident indicator. for example. changes in stock prices have been taken as providing information about the future course of the real economy. real wages. Finally. In the pre-WWI period and interwar period. hence. The behavior of prices and inﬂation appears to have changed over time. although it is important to emphasize that the market has fallen without a subsequent recession. inﬂation was procyclical with a very low mean. However. contemporaneous correlations between output ﬂuctuations in different countries were highest in the interwar period. Witness the impact of the large oil shocks in the 1970s. The stock market is positively related to the subsequent growth rate of real GDP. employment. prices and inﬂation were procyclical in the pre-WWII era. 16. Since the early 1980s. 18. inﬂation appears to be countercyclical. Between 1945 and 1980. Chadha and Nolan [62] identify the stylized facts
. the increased persistence of inﬂation and the countercyclical behavior of prices observed in this era. 17. 19. the stock market fell in the quarter before each of the eight recessions. The changes in the cyclical behavior of prices and inﬂation also have implications for the socalled impulses and propagation mechanisms of business cycles. In this sense. The standard deviation of inﬂation is lower than that of real GDP. a topic to which we will return. One way to rationalize this phenomenon may be in terms of monetary policy that is more accommodative — under a gold standard. and productivity have proved among the most difﬁcult to reconcile by current business cycle theories. Money (M2) is procyclical and tends to be a leading indicator of output.18
Business Cycles: Fact. reﬂecting the common experience of the Great Depression. By contrast. 20. A similar change appears to have characterized the behavior of price-level ﬂuctuations. The ﬁndings concerning the behavior of hours. 15. There is also a marked increase in the persistence of inﬂation after WWII. This set of facts constitutes a benchmark which a successful business cycle model should meet. Fallacy and Fantasy
14. the procyclicality of M2 has diminished since the 1980s.

Facts
19
of business cycles for the UK economy over a long sample period. which is independent of the US cycle. Artis. Christina Romer [180. If the higher-quality data in the post-WWII period are subjected to a transformation that makes the pre. the variability of output appears to have diminished in the post-WWII period. Artis and Zhang [18. 181] has argued that the ﬁnding of lower output variability in the postWWII period is due to changes in the measurement of output in the preand post-WWII eras.and post-WWII data of comparable quality. While her ﬁndings have garnered much interest. Kontolemis. They ﬁnd that the standard deviations of a typical variable in their sample declined by about 20–40% since the 1980s. A’Hearna and Woitek [2] establish the facts of business cycles in the late 19th century using spectral techniques. They suggest that if one considers additional evidence based on a larger sample of countries over the different periods. THE EURO AREA BUSINESS CYCLE
Much of the early work on the European business cycle was concerned with examining the impact of monetary union arrangements on economic activity. 19] investigate the relationship of the European Exchange Rate Mechanism (ERM) to the international business cycle in terms of the linkage and synchronization of cyclical ﬂuctuations between countries. Backus and Kehoe [23] argue that it is difﬁcult to reach a ﬁrm conclusion on this point based on US data alone. Their ﬁndings suggest the emergence of a group-speciﬁc European business cycle since the formation of the ERM. and another 15% to reduced shocks to food and commodity prices. They attribute around 20–30% of the reduction in output volatility to better monetary policy. Stock and Watson [198] seek to avoid the problem of poor data quality in the pre-WWII period. then one discerns no change in output ﬂuctuations across the two periods. On the one hand. and compare the volatility of 21 different variables in the 20 years since the 1980s and the 40 years in the period following WWII up to the 1980s. Their ﬁndings also provide evidence on the factors behind the “Great Moderation”.3. The above ﬁndings can also shed light on whether output ﬂuctuations have moderated in the post-WWII era. The countries selected are
. and Osborn [20] propose business cycle turning points for a number of countries based on industrial production.
2. 15% to smaller shocks to productivity.

and Proietti [21] discuss alternative approaches to dating euro area business cycles. and to determine whether cyclical movements are similar across different countries. Suppose that a peak terminates an expansion. Then. which implies that the probability of continuing an expansion pEE = Pr(ECt+1 |ECt ) is given by pEE = 1 − pEP .20
Business Cycles: Fact. Let St denote a discrete random variable that follows a ﬁrst-order Markov process
3This algorithm follows Harding and Pagan [115]. Whereas in the post-WWII era the euro area countries exhibited high rates of growth. conditional on being in a contraction. These authors also articulate a formal model of turning points based on restrictions imposed on a Markov process. who extended the Bry and Boschan [49] algorithm to a quarterly setting. the probability of continuing a contraction pRR = Pr(RCt+1 |RCt ) is pRR = 1 − pRT . Tt trough
Let pEP = Pr(Pt+1 |ECt ) denote the probability of transiting to a peak. and a trough terminates a recession. suppose that the economy can be in one of two mutually exclusive states. Marcellino.3 To describe this algorithm. The euro area experience makes consideration of both types of business cycles relevant. They use this information to examine the international nature of cyclical movements. expansion Et or recession Rt . they distinguish between classical and growth cycles. making growth cycles the relevant concept. They also consider the lead/lag relationships between countries at peaks and troughs. The Markov process distinguishes between turning points within the two states by assuming that Et = Rt = ECt expansion continuation . Fallacy and Fantasy
the G7 countries along with the prominent European countries. implying that classical cycles should also be considered. Artis. As in the Stock and Watson [197] approach. conditional on being in an expansion. Pt peak RCt recession continuation .
. Likewise. deﬁne pRT = Pr(Tt+1 |RCt ) as the probability of transiting to a trough. in recent years growth has slowed and even absolute declines in real GDP have been observed.

St ). as follows: ETSt = {( yt+1 < 0) ∪ ( RTSt = {( yt+1 > 0) ∪ (
2 2
yt+2 < 0)} yt+2 > 0)}. Now consider applying this algorithm to dating classical cycles. Tt ) belong to a contraction. the probability that the economy will transit to a trough. Pt ) both belong to an expansion and (RCt . St = ECt . if ETSt is false. yt+1 = yt+1 − yt < 0 and 2 yt+2 = yt+2 − yt < 0 represent the candidate sequence for terminating an expansion. St−1 . and a recession termination sequence. Then. is pRT . pEP . Likewise. This is the joint event that the history at time t. a peak at date t − 4. conditional on having been in an expansionary phase.
. Hence. yt+1 = yt+1 − yt > 0 and 2 yt+2 = yt+2 − yt > 0 represent the candidate sequence for terminating a contraction. Else. respectively. and on complete cycles. Pt+1 . the expansion continues. Let yt denote the underlying series. can only be followed by a peak at t + 1. which is automatically satisﬁed since the states (ECt . Likewise. St−2 . RTSt .
Thus. St−2 .Facts
21
with four states and transition probability matrix as follows:
ECt+1 ECt Pt RCt Tt pEE Pt+1 pEP RCt+1 Tt+1
0 0 1
0 0 0
0 1
pRR
0 0
pRT
0
0
The dating rules impose minimum durations on a phase. expansion or contraction. we can deﬁne an expansion termination sequence. The duration of a phase is required to be two quarters. St−1 . Pt−4 . peak-to-peak or trough-to-trough. St−3 . A similar condition exists for troughs. which deﬁnes a trough. This is the joint event that the history at time t. St = RCt . and that ETSt is true. St ). ETSt . St−3 . which deﬁnes a peak. (St−4 . (St−4 . The algorithm is completed by ﬁnding the probability that the economy will transit to a peak. features an expansionary state at time t. The minimum duration of a complete cycle is given as ﬁve quarters. conditional on having been in a contractionary phase. features a contractionary state at time t.

Euro Area Business Cycle Turning Points. Marcellino. Otherwise.4 Table 2.22
Business Cycles: Fact.
. Henry. Fallacy and Fantasy
and that RTSt is true. and Mestre [85].
Trough 1975 (I) 1981 (I) 1982 (IV) 1993 (I) 1975 (III) 1977 (IV) 1982 (IV) 1993 (II) 1997 (I)
1975 (I) 1982 (IIII) 1993 (III)
4 See Fagan. Its stated mission is to establish the chronology of recessions and expansions of the 11 original euro area member countries for 1970–1998 and of the current euro area as a whole since 1999. The turning points determined by the CEPR Business Cycle Dating Committee are also indicated in this table. Peak (1) Classical Cycles∗ 1974 (III) 1980 (I) 1982 (II) 1992 (I) (2) Deviation Cycles∗ 1974 (I) 1976 (IV) 1980 (I) 1990 (II) 1995 (I) 1998 (I) (3) CEPR Dating Committee 1974 (III) 1980 (I) 1992 (I)
∗ Artis. [21] also examine the properties of alternative ﬁlters such as the Hodrick–Prescott and Baxter–King ﬁlters for decomposing a time series into trend and cyclical components. Artis et al. and a trough cannot occur if output is above it. if RTSt is false. We note that these are similar to the classical cycles identiﬁed by Artis et al. [21]. The dating of deviation cycles is achieved by modifying this algorithm to account for the fact that a peak cannot occur if output is below trend. and Proietti [21].
Table 2. They implement their procedures using data on euro area GDP and its components for the European Central Bank’s (ECB) Area-Wide Model for the period 1970–2001. the contraction continues. Finally.2 shows the business cycle turning points for both classical and deviation or growth cycles obtained using this approach.2. The Centre for Economic Policy Research (CEPR) has recently formed a committee to set the dates of the euro area business cycle.

the eurozone countries and English-speaking countries (including Canada. and Terrones [139] and others.0 0.4 0. France.00 13. Prasad. Stock and Watson [199] also seek to provide evidence on the sources of the changes. which is speciﬁed in terms of the growth rates of quarterly GDP for the G7 countries. They base their results on various measures of correlation of GDP growth across countries. This is a standard structural vector autoregression (VAR) with an unobserved factor structure imposed on the VAR innovations. 2. UK. Stock and Watson [199] provide a comprehensive analysis of the volatility and persistence of business cycles in G7 countries (deﬁned to include the US. Italy. they ﬁnd no evidence for closer international synchronization over their period of study.8 -0. This is similar to the ﬁndings of Köse.
Fig. 2. and Canada) over the period 1960– 2002. Japan. They ﬁnd that the volatility of business cycles has moderated in most G7 countries over the past 40 years. and the US). they ﬁnd evidence on the emergence of two cyclically coherent groups. They also provide evidence on the synchronization of international business cycles.3.01 14. namely. Germany. they use a socalled factor-structural vector autoregression (FSVAR).02 -0.Facts
14.4 1970
1975
1980
1985
1990
1995
2000
2005
1975
1980
1985
1990
1995
2000
2005
BPTRENDEURO
EUROAREA
BPCYCLEEURO
Fig. the UK.03 1970
23
14.02 0.3 illustrates the logarithm of real GDP for the euro area and its trend and cyclical components estimated according to the Baxter–King ﬁlter for the period 1970(I)–2005(III). do they arise from changes in the magnitudes of the shocks or the nature of the propagation mechanism? Are the sources of the changes domestic or international? To answer these questions. Trend and Cyclical Components in Euro Area GDP. Let Yt denote the n × 1 vector of
.2
13. First.6 -0. in sync with Artis et al. However.01 13. [20].03 0.

. However. . for example. . . Notice that the common shocks affect the output of multiple countries contemporaneously. . Fallacy and Fantasy
real GDP growth for the G7 countries considered in this study. and Germany.5)
where ft is a k × 1 vector with k ≤ n. the elements of t denote the country-speciﬁc idiosyncratic shocks. They also provide a variance decomposition for the impact of (i) the common international shocks. and the least for Italy and Japan. the role of international sources of ﬂuctuations arising from common shocks or from
. The covariance structure of the shocks is given by E (ft ft ) = diag (σf 1 . This provides the identiﬁcation scheme to help identify the common international shocks. First.24
Business Cycles: Fact. σfk ) and E (νt νt ) = diag (σν1 . Second. Stock and Watson [199] ﬁnd evidence for two common shocks. The model allows for spillover effects to occur through the lagged effect of an idiosyncratic shock. whereas the idiosyncratic shocks affect them with a lag. In the ﬁrst period. . σνk ). their relative variance varies by country and by time period.6) (3. and (iii) the spillover effects of the domestic shocks on the h-step-ahead forecast error for each country. The standard VAR model is given by Yt = A(L)Yt−1 + νt . (3. the impact of international shocks is estimated to be greatest for countries such as Canada. These may arise from the role of international trade. and the elements of ft denote the common international shocks. . In the second period. domestic shocks explain almost all of the variance for Japan. (ii) the domestic shocks. France. where the elements of ft and t are assumed to be mutually independent. They consider two sample periods: 1960–1983 and 1984–2002. they ﬁnd that most of the variance of GDP growth can be attributed to common and idiosyncratic domestic shocks. . where the vector of reduced-form errors vt has the factor structure ν t = ft + t . In this representation. reﬂecting the impact of the ten-year-long deﬂationary episode for this country.

Head.t = ai + biworld ft world + bi
region region fr. Their overall results indicate that the magnitudes of common international shocks appear to have become smaller in the 1980s and 1990s than they were in the 1960s and 1970s. consumption. where r stands for North America.4. France. • the single world factor ( f world ).
(4. Canada. or Oceania). Latin America. IS THERE A WORLD BUSINESS CYCLE?
Köse. for example. Let N denote the number of countries. and Whiteman [138] consider the issue of a world business cycle. and the UK. . . Gregory. Germany.t denote the observable variables for i = 1. . . they also ﬁnd that shocks have become more persistent in countries such as Canada.
. they also show that the decline in overall volatility of GDP growth for countries such as the US. . They argue that many recent studies of international business cycles focus on a subset of countries — not the world.5 Their approach assumes that there are K unobservable factors that are hypothesized to characterize the dynamic interrelationship among a cross-country set of economic time series. and Italy. region . . Otrok. and T the number of time periods. one per each country). • R regional factors ( fr Africa.7)
5 For recent studies. see. Finally. T . Developing Asia. Developed Asia. . However. Let yi. N × M . t = 1. and the UK is due to the decline in the variance arising from international shocks. . and use data on 60-odd countries covering seven regions of the world to determine common factors underlying the cyclical ﬂuctuations in the main macroeconomic aggregates (output. and also of the failure of business cycles to become more synchronized despite the great increase in trade ﬂows over this period.
2.Facts
25
spillovers appears to have increased for the US. and Raynauld [109] or Lumsdaine and Prasad [154].t
+
i. M the number of time series per country.t . There are three types of factors: • N country-speciﬁc factors ( fn . This is the source of the moderation of individual business cycles. Europe. and investment) across all countries and regions as well as across countries and aggregates separately. The behavior of each observable variable is related to the factors as follows: yi.t country
+ bi
country country fn.

t
= φi. Fallacy and Fantasy
j
where E ( i. There are k several ways to estimate this model. . where N + R + 1 < MN . Thus. M × N . Let fk. the sign of the factor loading for US output is considered positive. . one approach is to use a Kalman ﬁltering algorithm together with classical statistical techniques to estimate the model’s parameters.t ) = σf2 .t is due to the common factors. and the country factors are identiﬁed by positive factor loadings on the output of each country.
(4.2
i. and to follow autoregressions of orders pi and qk . .1
i. K . .t−pi
+ ui. Since the factors are unobservable. and zero otherwise.t can be explained by each factor. In what follows.t−2
+ · · · + φfk . which themselves may be serially correlated. scales are identiﬁed by assuming that each σf2 is equal to a constant. k = 1. the model assumes that the behavior of M × N time series can be explained by N + R + 1 factors.t−1
+ φfk .qk
fk .t−1
+ φi.2
fk . Likewise. E (ufk . This procedure yields a joint posterior distribution for the unobserved factors and the unknown parameters. and ufk .t fk . .
(4. i. The innovations k ui.t ufk . The coefﬁcients bi denote the factor loadings.8)
where E (ui. . contemporaneously uncorrelated random variables.t . the sign of the factor loading corresponding to North America is considered positive.t . .t . are mean zero.t ui.t . an alternative is to use Bayesian estimation techniques. conditional on the data. Thus.
fk .t = Then. One identiﬁcation device is to assume that the factor loading on one of the factors is positive. and defer a discussion of the estimation techniques to Chapter 7. Since the common factors are unobserved. as
i. all the covariation among the observable series yi.t−2
+ · · · + φi.t−s ) = σi2 for i = j and s = 0. we report results based on the median of the posterior distribution for the factors. For the world factor.1
fk .t ui.
(4. s.10)
where E (ufk . .26
Business Cycles: Fact. Both the idiosyncratic errors and the factors are allowed to be serially correlated. for the regional factor.t−qk
+ ufk . we cannot identify uniquely the sign or scale of the factors or the factor loadings.t−s ) = 0 for all k. Finally. i = 1.9)
= φfk . and show how much of the variation in yi.pi
i.t j.
.t ) = 0 for i = j.t . respectively.

and the downturn and recession of the early 1990s. By contrast. This ﬁnding has been documented further in the international business cycle literature.t ). the recession of the 1980s associated with the debt crisis in developing countries and the tight monetary policies in the major developed countries. and country factors are modeled as autoregressive processes which may display substantial persistence depending on the estimated values of the parameters. Third. the world factor is more successful in explaining economic activity in developed relative to developing countries. regional. the regional and country factors explain covariation or co-movement that is less persistent. Speciﬁcally. we have that Var(yi. Var(yi. + (bi
Thus. 9% of consumption growth. A second important ﬁnding from the paper is that most of the persistent co-movement across countries and aggregates is captured by the world factor. ﬁrst.t ) They ﬁnd evidence that. the world factor based on a large set of developed and developing countries implies that the recession of the 1980s was. the world factor explains nearly 15% of variation in output growth. they ﬁnd that country and idiosyncratic factors account for a much larger share of variability in investment growth than the world and regional factors. and 7% of investment growth. Using the representation of the model. in contrast to models estimated using a smaller sample of countries. The paper also allows for a simple decomposition of variance attributed to the different factors.t
region
) (4. They interpret this ﬁnding as evidence for a world business cycle. However. as severe as the recession of the mid-1970s. the fraction of variance attributed to the world factor can be expressed as (biworld )2 Var(ft world ) . if anything. the recession of the 1970s associated with the ﬁrst oil shock.Facts
27
Köse et al. country factors account for a much larger fraction of consumption growth than output growth. which is discussed in detail in Chapter 4. Second.’s paper [138] shows that the world factor identiﬁes many of the major cyclical events over a 30-year period: the expansionary periods of the 1960s.11)
country 2 country ) Var(fn. Recall that the world. Second.t ) + Var( i.t ) = (biworld )2 Var(ft world ) + (bi
region 2
) Var(fr. The authors also document some noteworthy ﬁndings regarding the sources of volatility of investment growth. idiosyncratic
.

Rodrik [178] emphasizes the importance of political risk (in his case. prices. The last two ﬁndings of the paper imply that regional factors play a minor role in aggregate output ﬂuctuations. • 1870–1941: This era corresponded to the classic gold standard. and Demers [10] argue that an important source of risk for developing countries is political risk or risk arising from the threat of expropriation. The authors pose this as a puzzle. Fallacy and Fantasy
shocks explain a much larger fraction of the volatility in investment growth in developing countries relative to developed countries. the UK.7 We conjecture that variability in investment due to such factors is a key channel for inducing idiosyncratic volatility in developing economies’ investment behavior. However. They divide this period into four periods that also reﬂect the monetary and capital account regimes prevailing in them. Yet this ﬁnding could also be due to model misspeciﬁcation.5. 7 In a related literature.
6 For recent reviews. Several studies document the importance of political risk in the unsuccessful recovery of private investment following the adoption of IMF stabilization packages in various countries. the dynamic factor model suffers from the shortcoming that all covariation is attributed to the common factors. there is also the possibility that some observed covariation could result from the spillover effects of idiosyncratic or country shocks.
. HISTORICAL BUSINESS CYCLES
Basu and Taylor [31] examine business cycles within an international historical perspective. much work on investment behavior shows that irreversible investment decisions are particularly susceptible to risk and uncertainty. and debt and currency crises. This last result is interpreted as evidence against the existence of a European business cycle. for example. especially in an era of increased trade ﬂows and ﬁnancial integration. Yet. exchange rates.6 Altug. Paralleling this ﬁnding. the risk of policy reversal) for irreversible investment decisions. It featured ﬁxed exchange rates and worldwide capital market integration. As Stock and Watson [199] note. Demers.
2.28
Business Cycles: Fact. there is little evidence that the volatility of European aggregates can be attributed to the European regional factor. and other European countries plus Argentina for the period since 1870. See. Demers. Serven and Solimano [189]. real wages. disruptions to market access. and Altug [79]. They consider the time series behavior of output. see Caballero [53] or Demers. policy reversals. investment. total consumption. and the current account for 15 countries including the US.

the so-called concordance correlation. which examines whether the turning points in the different series occur at similar dates. Bordo and Heibling [44] ask whether national business cycles have become more synchronized. the world economy shifted from a globalized regime to an autarkic regime. This period also corresponded to the Great Depression. Norway. Early 1970s–present: This period features a ﬂoating exchange regime and a period of increasing globalization.
. which corresponded to the post-World War II era of reconstruction and a resumption of global trade and capital ﬂows. Denmark. The countries that they examine are Australia. Sweden. 1945–1971: The third regime is the Bretton Woods era. For example. the UK. Italy. which simultaneously occurred in a number of countries. They also use standard output correlations and factor-based measures. thus allows for an examination of such mechanisms as nominal rigidities in propagating shocks. and the US. and consider the four exchange rate regimes also examined by Basu and Taylor [31].versus supply-side factors or shocks and the role of alternative propagation mechanisms such as price rigidity. the Netherlands. Switzerland. France. Consideration of alternative historical periods. most explanations of the Great Depression attribute the source of this massive downturn. they use a statistical measure of synchronization developed by Harding and Pagan [115]. Germany.” Among other concepts. Spain. Portugal. Bordo and Heibling [44] use real GDP or industrial production series to determine synchronization. Canada. They examine business cycles in 16 countries during the period 1880–2001. to monetary phenomena. Basu and Taylor [31] also argue that their periodization allows for an analysis of international capital ﬂows and capital mobility in affecting cyclical ﬂuctuations. Japan.
Basu and Taylor [31] argue that considering such a breakdown allows for the analysis of the impact of different regimes on cyclical phenomena. Synchronization refers to the notion that “the timing and magnitudes of major changes in economic activity appear increasingly similar. Finland.Facts
29
•
•
•
1919–1939: During this period. Instead of using information on NBER-type reference dates. including the era of the Great Depression. It also provides a way to identify the importance of demand.

Note ∗ that the variance of Iij is given by ∗ Var(Iij )
4 = 2E T
T t=1
¯ ¯ (Sit − Si )(Sjt − Sj ) . Bordo and Heibling [44] calculate the business cycle indicators Sit for each of the four exchange rate regimes. and if the two cycles are unrelated. they ﬁnd that the average of the correlation coefﬁcients is zero. say. In the interwar and post-Bretton Woods era.12)
¯ Let Si = (1/T ) T Sit denote the estimated probability of being in an t=1 expansion.
If the two cycles are perfectly synchronized (so that Sit = Sjt for all t). if they are in different states in each period (so that Sit = 1 − Sjt ).
(5. Subtracting this from Iij yields the mean-corrected concordance index:
∗ Iij
1 =2 T
T t=1
¯ ¯ (Sit − Si )(Sjt − Sj ). For the gold standard era.30
Business Cycles: Fact. we divide Iij by a consistent estimate of its standard error under the null hypothesis of independence. cycles were. let Sit and Sjt denote binary-cycle indicator variables which assume a value of 1 if the economy is in an expansion. They deﬁne a recession as one or more consecutive years of real GDP growth. the standardized index equals 0. then the standardized index equals −1. then the standardized index equals 1. Fallacy and Fantasy
To brieﬂy describe the concordance correlations. The key ﬁnding is that during the classical gold standard. for countries i and j. as half of all pairs of business cycles are negatively related to each other while the other half are positively related. more than half of all national business cycles become positively related to each other. then under the assumption that Sit and Sjt are independent. uncorrelated
. ¯ ¯ ¯ ¯ the expected value of the concordance index is 2Si Sj + 1 − Si − Sj . 0 otherwise. They argue that Sit = 1 for most countries during the Bretton Woods era of 1948–1972 and hence leave this period out.
t=1
(5. while an expansion is deﬁned as one or more years of positive GDP growth.13)
∗ To derive the concordance correlation coefﬁcient. on average. A simple measure of concordance is deﬁned by the variable 1 Iij = T
T
[Sit Sjt − (1 − Sit )(1 − Sjt )].

as in Stock and Watson [199]. more positive bilateral output correlations by era. and a trade linkages version. and that the increasing importance of global shocks reﬂects the forces of globalization. such crises have ﬁgured importantly in emerging market output ﬂuctuations. and the idiosyncratic shocks are allowed to be serially correlated and heteroscedastic. They consider a so-called center country version of this model. lagged GDP growth of the center country is included alongside the lagged value of the own country’s GDP growth in the VAR representation. In the trade linkages version. and spillover effects in generating this result. they started becoming synchronized with each other. before 1914 and after 1971. The authors also estimate a so-called static approximate factor model for the growth rates of GDP.Facts
31
with each other. In the former. suggesting that it is not transmission from the center country that accounts for the increased role of transmission. idiosyncratic shocks. Finally. They argue that banking crises were less severe in the period before 1915.” They also estimate restricted versions of a FSVAR model along the lines of Stock and Watson [199] to determine the role of common shocks.
. but this was not so for ﬁnancial or twin crises. there are no dynamics in the relation between output growth rates and the factors. However. They ﬁnd that both global (common) shocks and transmission have become more important. The authors also examine standard output correlations. Typically. especially the integration of goods and services through international trade and the integration of ﬁnancial markets. They ﬁnd that there has been a tendency for higher. they ﬁnd an increase in correlations for the Anglo-Saxon countries. In this model. the importance of transmission for peripheral countries arises only in the trade model. They also ﬁnd higher output correlations for the core European countries (the EEC) and the Continental European countries. whereas beginning with the interwar period. lagged GDP growth of the major trading partner is included in place of the center country’s. Eichengreen and Bordo [84] examine the nature of crises in two periods of globalization. which measure the magnitude as well as the direction of output changes. They ﬁnd that the variability of output due to variability in the common factors has “doubled from about 20 percent during the Gold Standard era to about 40 percent during the modern era of ﬂexible exchange rates. We discuss emerging market business cycles in detail in Chapter 6. Bordo and Heibling [44] conclude by noting that global shocks are the dominant inﬂuence across all regimes.

This page intentionally left blank
.

changes in their subjective beliefs could help to trigger business cycles. the result is unemployment and greater declines in output relative to a situation with ﬂexible prices. In a simple labor market model. Much of the controversy in the business cycle literature has stemmed from differences attached to the importance of alternative propagation mechanisms and the associated shocks.Chapter 3
Models of Business Cycles
The standard approach to classifying business cycle models follows Ragnar Frisch’s [95] terminology of impulses and propagation mechanisms. The shocks or impulses thought to instigate business cycles have typically been varied and diverse. Keynesian and New Keynesian models stress the role of frictions such as price stickiness. more precisely. as Keynes [127] did. one could also assign a role to taste shocks or changes in preferences of consumers. whether they are permanent or transitory such as oil shocks. wars. This approach emphasizes the role of intertemporal substitution motives in propagating shocks. The Great
33
. Weather shocks have always had a place among the impulses thought to trigger ﬂuctuations in economic activity. More controversially. Technology shocks which alter a society’s production possibilities frontier. one could argue. Current real business cycle (RBC) theory argues that business cycles can arise in frictionless. By contrast. have ﬁgured prominently in the recent literature. complete markets in which there are real or technology shocks. and other disruptions in market activity have also been acknowledged to play a role. that investors’ “animal spirits” or. perfectly competitive. Political shocks. Equivalently. if real wages do not adjust downward when there is a negative shock to demand.

Some of the extensions of the RBC approach have modiﬁed the basic framework to incorporate the role of government. shocks originating in various ﬁnancial markets lead to a widening circle of bankruptcies and bank failures as well as large and negative effects on real output. 173). Bernanke and Gertler [40] or Kiyotaki and Moore [134].
1 See.
. All markets are perfectly competitive. and international trade. As Kydland and Prescott argue. and all prices adjust instantaneously to clear markets. we discuss variants of the RBC model. and a constant-returns-to-scale (CRTS) production technology for producing the single good using labor and capital. In the following chapter. we discuss an international RBC model to describe how the transmission mechanisms proposed by this approach translate into an open-economy setting.2
3. this is a crucial modiﬁcation as “more than half of business cycle ﬂuctuations are accounted for by variations in the labor input” ([143]. There is a representative consumer who derives utility from consumption and leisure. 2 Rebelo [177] provides a recent review of alternative models and directions associated with the RBC agenda. and that there is no trade with the rest of the world. that households consume only goods produced in the market. The prototypical RBC model has the structure of a standard neoclassical growth model with a labor/leisure choice incorporated. all factors are fully employed. Fallacy and Fantasy
Depression and the recent ﬁnancial crises also indicate the importance of credit market frictions and ﬁnancial accelerator-type effects as a potential propagation mechanism for business cycles. In this case. We will discuss these extensions in the following sections. A number of these assumptions have come under criticism in the recent literature. for example. Since the debate has revolved around the efﬁcacy of technology shocks and intertemporal substitution effects in replicating business cycles. The standard RBC model also assumes that there is no government. p.1 In this chapter.1. but his discussion is abbreviated to satisfy the constraints imposed by a journal article. AN RBC MODEL
RBC theory has become popular in recent years because its micro foundations are fully speciﬁed and it links the short run with the neoclassical growth model.34
Business Cycles: Fact. home production. it seems imperative to discuss the main propagation mechanisms of the RBC model.

and N (0.1)
where 0 < β < 1 is the discount factor. (1.d. We will discuss the propagation mechanism when describing the solution for the model.. 0 < θ < 1. zt+1 = µ+ρzt + 0 < α < 1. we will suppose that the utility function has the form U (ct .2)
Hence.
(1.
. lt ) . (1. The aggregate feasibility constraint is deﬁned as ct + it = yt . There is a representative ﬁrm with CRTS production that is affected by a stochastic technology shock each period: yt = exp (zt )ktα ht1−α .
t
∼ i.3 Assuming a solution exists. Altug and Labadie [6] or Stokey and Lucas with Prescott [201].i. For ease of exposition. for example.Models of Business Cycles
35
Turning to the basic model.5)
where it denotes economy-wide investment and 0 < δ < 1 is the depreciation rate. lt ) = ctθ lt1−θ . time spent not working is taken as leisure.
0 < ρ < 1. (1.
The time constraint requires that the sum of leisure and labor hours equals 1: lt + ht = 1. which is assumed to be stationary but persistent as long as ρ = 1. The problem described above is an inﬁnite-horizon dynamic stochastic optimization problem.6)
Notice that the only shock in this model is the technology shock. σ 2 ). The existence of a solution can be shown using standard recursive methods for analyzing such problems.
(1. (1.
3 See.4)
Capital evolves according to kt+1 = (1 − δ)kt + it . the representative agent has time-separable preferences over consumption and leisure choices given by
∞
U = E0
t=0
βt u(ct .3)
Assume that the technology shock follows an AR(1) process:
t+1 .

kt+1
(1. zt+1 ) = λt . Substituting for lt = 1−ht using the time constraint.9) (1. βEt Vk (kt+1 . his choice is between how much to work and how much time to spend in leisure.36
Business Cycles: Fact. Notice that this is just a simple Robinson Crusoe economy in which the Crusoe produces output using capital and labor. lt ) = λt exp (zt )(1 − α)ktα ht−α .13)
The ﬁrst equation shows that the marginal rate of substitution between consumption and leisure is equal to the marginal product of labor. U1t (1.lt . The second equation is the intertemporal Euler equation. zt ) = λt exp (zt )αktα−1 ht1−α + (1 − δ) . Thus. This means that the capital accumulation process will also be a function of the capital-labor ratio. (1. To analyze the optimal consumption-leisure
. respectively.10) (1.8) (1. ¯ it = 0. which he then consumes. Observe that the ratio U2 /U1 is a function of the capital-labor ratio. Let λt denote the Lagrange multiplier on the resource constraint. the ﬁrst-order conditions with respect to ct . kt = k. lt + ht = 1. kt+1 and lt and the envelope condition are U1 (ct . lt ) = λt . These conditions can be rewritten as U2t = exp (zt )(1 − α)(kt /ht )α U1t βEt U1. There is no unemployment in the model as time not spent working is (optimally) taken as leisure.7)
subject to ct + kt+1 = exp (zt )ktα ht1−α + (1 − δ)kt . that is. zt ) = max {U (ct . U2 (ct . lt ) + βEt V (kt+1 . Vk (kt .11)
where U1t and U2t denote the marginal utility of consumption and leisure.t+1 [exp (zt+1 )α(kt+1 /ht+1 )α−1 + (1 − δ)] = 1. and ct = yt .12) (1. Fallacy and Fantasy
let the value function associated with the problem be expressed as V (kt . zt+1 )}
ct . kt /ht . Suppose for the moment that there is no investment in the model.

Notice that the technology shock does not affect optimal hours worked because. (1. we have the intertemporal Euler equation characterizing the behavior of the optimal capital stock over time. for this preference speciﬁcation. Then. Figure 3.15)
Thus.
. we can rewrite the intertemporal Euler equation by noting that U1 (ct+i . for example. θ(1 − ht ) We can solve this equation for the optimal labor hours ht∗ as ht∗ = θ(1 − α) . Now suppose output can be consumed or invested so that it = 0. if θ = 2/3 and α = 1/2. Using our preference speciﬁcation. then Crusoe spends 50% of his time working and 50% taking leisure.14). Thus. However. in addition to equation (1. This yields βEt exp (zt+1 ) kt+1 ht+1
α θ−1
α exp (zt+1 )
kt+1 ht+1
α−1
+1−δ
kt = exp (zt ) ht
α θ−1
. 1 − θ + θα (1. Crusoe will typically respond to a temporary negative shock by lowering savings and also by working less.1 describes the representative agent’s optimum.14) θlt Substituting for ct = yt = exp (zt )ktα ht1−α yields (1 − θ) exp (zt )ktα ht1−α = exp (zt )(1 − α)(kt /ht )α . we note that if the substitution effect of an increase in productivity dominates the income effect. then Crusoe will work less in response to a temporary negative technology shock and will also consume and produce less.14).Models of Business Cycles
37
choice. the income and substitution effects of an increase in productivity on hours worked cancel each other out. we consider equation (1.12).16)
Now a temporary negative technology shock also affects investment or saving.
(1. which under our preference speciﬁcation becomes (1 − θ)ct = exp (zt )(1 − α)(kt /ht )α . lt+i ) = θ(ct+i /lt+i )θ−1 for i ≥ 0 and using the result in equation (1. the impact of these changes is to induce ﬂuctuations in labor supply and employment as well as investment.

it will have a lower capital stock in the future. if the economy experiences a period with lower investment. Under constant returns to scale.
To derive an observable measure of the Solow residual. To derive the Solow residual.3) and then take the ﬁrst differences: ln (yt+1 ) = zt+1 + α ln (kt+1 ) + (1 − α) ln (ht+1 ). 3. skt + sht = 1. the parameter α is typically measured as the share of capital in real output deﬁned as skt = rt kt /pt yt .1. the growth in the total factor productivity (TFP) is measured as a residual: zt+1 = ln (yt+1 ) − skt ln (kt+1 ) − (1 − skt ) ln (ht+1 ). Under these assumptions. implying that the effect of the shock persists. where sht = wt ht /pt yt denotes the share of labor in national income.
Furthermore.
. where rt denotes the competitively determined rental rate on capital and pt denotes the product price normalized here as unity.
Robinson Crusoe’s Optimum. We can also derive a measure of the Solow residual for this model by using the form of the production function.38
Business Cycles: Fact. Fallacy and Fantasy
Fig. take the logarithm of both sides of equation (1.

we can substitute for consumption in the utility function to obtain u(ct ) = u( exp (zt )ktα ht1−α − it . We implement a quadratic approximation procedure that was initially suggested by Kydland and Prescott [141]. according to Denison [80].Models of Business Cycles
39
Solow [196] showed that the residual accounted for about one half of the US GDP growth between 1909 and 1949.
t+1 . Step 1: Compute the deterministic steady state for the model.4).14) and (1.
The quadratic approximation procedure is implemented as follows.19)
0 ≤ ht ≤ 1. 40% of GNP growth in the USA between 1929 and 1957 resulted from technical development. the problem now becomes
∞ {it . Notice that in the RBC framework the Solow residual is typically treated as exogenous. Letting u(ct . Critics of the RBC approach argue that there are endogenous components to the ﬂuctuations in the Solow residual.ht }t=0
max E0 ∞
t=0
θ ln ( exp (zt )ktα ht1−α − it ) + (1 − θ) ln (1 − ht )
subject to kt+1 = (1 − δ)kt + it .2.
3. Similarly. A NUMERICAL SOLUTION
We now provide a numerical solution to further illustrate the properties of the standard RBC model. We assume that capital depreciates at the rate 0 < δ < 1 and the technology shock follows the process in (1.
(2. and convert the original nonlinear dynamic optimization procedure to one which has a quadratic objective and linear constraints (see also Christiano [64]).16) together with the feasibility
. consumption plus investment equals output at the optimum. rendering the RBC interpretation invalid. This is obtained by setting the exogenous technology shock equal to its mean value and evaluating the conditions (1. This is an issue that has been contested in the business cycle literature. zt+1 = µ + ρzt + ct ≥ 0. 0 < θ < 1. 1 − ht ) = θ ln (ct ) + (1 − θ) ln (1 − ht ). Since the utility function is strictly increasing. Using this fact.18) (2. 1 − h).17) (2.

(2. the solution for the stochastic optimal control problem is identical to the solution for the deterministic version of the problem with the shocks εt+1 replaced by their expectation E (εt+1 ). Substituting for next period’s state variables and using this expression for the value function. and ut . notice that the value function will be a quadratic function in the state variables: V (x) = x Px + d . in other words.42
Business Cycles: Fact. t ≥ 0. This class of problems is known as optimal linear regulator problems. xt . and Sargent [16] provide further discussion of the formulation and estimation of linear dynamic economic models. Bellman’s equation becomes x Px + d = max x Rx + u Qu + 2x Wu
u
+ βE (Ax + Bu + ε) P(Ax + Bu + ε) + d
. Ljungqvist and Sargent [146] and Anderson. the dynamic optimization problem can now be written as
∞ {ut }t=0
max E0 ∞
t=0
βt [xt Rxt + ut Qut + 2xt Wut ]
(2.24)
subject to the linear law of motion xt+1 = Axt + But + εt+1 . Such problems can be solved using the methods for solving dynamic optimization problems that satisfy a certainty equivalence property.26)
. (2. Given the structure of the problem. where d and P are quantities to be determined. Fallacy and Fantasy
Step 3: Convert the original dynamic optimization problem into a problem with linear constraints and a quadratic objective function. Hansen. McGrattan.25)
where E (εt+1 ) = 0 and E (εt εt ) = . This is now an optimal control problem with a quadratic objective and linear constraints. Using the deﬁnition of T . Bellman’s equation for this problem is given by V (xt ) = max xt Rxt + ut Qut + 2ut Wxt + βE [V (xt+1 )]
ut
subject to xt+1 = Axt +But +εt+1 .

A stochastic process for zt is also speciﬁed and a random number generator is used to simulate the TFP time series. A. where F = (Q + βB PB)−1 (βB PA + W ).
5To derive this result. and B must be further restricted to ensure the existence of a solution to the matrix difference equation deﬁning Pn . In the RBC literature. Substituting for u back into the deﬁnition of V (x) yields P = R + βA PA − (βA PB + W )(Q + βB PB)−1 (βB PA + W ).29) (2.5 Solving for u yields u = −(Q + βB PB)−1 (W x + βB PAx) = −Fx. (2.6 The solution for P can be obtained by iterating on the matrix Riccati difference equation: Pn+1 = R + βA Pn A − (βA Pn B + W )(Q + βB Pn B)−1 (βB Pn A + W ). starting from P0 = 0. output. Q .
. we have used the rules for differentiating quadratic forms as
∂u Qu ∂x Tu ∂u Tx = [Q + Q ]u = 2Qu.27)
Notice that the shocks εt+1 do not affect the optimal choice of ut . = T x and = Tx. given particular functional forms for the production function and the utility function (see Cooley and Prescott [75]). The equation for P is known as the algebraic matrix Riccati equation for so-called optimal linear regulator problems. The simulated series are used to generate time series for consumption. The policy function associated with Pn is Fn+1 = (Q + βB Pn B)−1 (βB Pn A + W ). the model is then calibrated with the data. ∂u ∂u ∂u 6 Notice that the matrices R. One condition that sufﬁces is that the eigenvalues of A are bounded in modulus below unity.Models of Business Cycles
43
The ﬁrst-order conditions with respect to u are Qu + W x + β[B PAx + B PBu] = 0.28) (2. d = (1 − β)−1 [βE (ε Pε)]. Calibration refers to the practice of determining the parameters of the model based on its steady state properties and the results of other studies.

By contrast. Fallacy and Fantasy
investment. so the approach differs signiﬁcantly from standard econometrics. variances. The steady values for the variables are given by k = 1.1134 12. ρ = 0. We then use the linear decision rules for all variables to simulate for the endogenous variables based on the same history of shocks. and cross-correlations for the simulated time series are compared to the comparable statistics in the data.9238 1. Cyclical Properties of Key Variables. and labor. productivity is almost as variable as output.028. the model is capable of delivering some of the salient features
Table 3.2952. covariances. δ = 0. The model is then judged to be close or not close based on this comparison.028. Standard Deviation (%) Output Consumption Investment Hours Productivity 8. and generate a sequence of ˆ technology shocks beginning from some initial value z0 as zt+1 = ρzt + ˆ t+1 .4934 0. with hours showing the least procyclicality. and ¯ σ = 0.1. In terms of the correlation of each series with output.95. Likewise. we ﬁnd that the variation in hours is typically quite low.9807 0.10. Consider the parameter values β = 0.6684 7. Speciﬁcally. Finally. we draw a sequence of shocks {ˆ t+1 }2999 that are normally distributed t=0 with mean zero and standard deviation 0. The typical set of unconditional moments used in the RBC literature is displayed in Table 3.44
Business Cycles: Fact.8616 8.95. Trends in the data are removed using a given ﬁlter. The means.1128. These are obtained by simulating the behavior of the different series across a given history of the shock sequence.9163 0.7496 Correlation with Output 1 0.3655.36. The RBC theorists have interpreted these results as implying that. in the ﬁrst instance. We now illustrate this solution method for the standard RBC model with a labor-leisure choice presented at the beginning of this section. α = 1/3.1. and h = 0.9840
.4783. We ﬁrst calculate a i ¯ ¯ set of unconditional moments that have been used in the RBC literature to match the model with the data. ¯ ¯ = 0. θ = 0. y = 0. we calculate unconditional moments for the different series after dropping the ﬁrst 1000 observations.1281. c = 0. We ﬁnd that consumption ﬂuctuates slightly less than output and investment ﬂuctuates signiﬁcantly more. we ﬁnd that all series are procyclical.

the technology shock explains much but not all of the variation in output. Consumption shows a gradual positive response because investment is initially high.1 1 % deviations form steady state 0. We note that hours and output both increase and then fall back to a lower level.8 0.Models of Business Cycles
45
of the data. By contrast. In terms of the variation accounted by the technology shocks.5 0.2.4
0
5
10 Years after shock
15
20
Fig. capital and consumption show a humped-shaped response.9 0. rising ﬁrst and then declining back to a lower level.
. as we observe in the data. These responses have been widely
Hours Output Consumption Capital Technology
1. Impulse Responses to a Shock in Technology.7 0. Consumption. Thus.2 illustrates the response of all the variables to a 1% shock in technology starting from the steady-state capital stock. the model predicts that consumption ﬂuctuates less than output whereas investment ﬂuctuates more. Figure 3. and productivity are all strongly procyclical. The percentage change in output exceeds the percentage change in the technology shock because optimal hours also show a positive response to the technology shock. 3. the RBC model has been viewed as being able to generate cycles endogenously and having economic signiﬁcance even if the calibration method is controversial.6 0. Speciﬁcally. investment. This is discussed further in the following sections.

46

Business Cycles: Fact, Fallacy and Fantasy

documented in the RBC literature (see, for example, Uhlig [206]). The positive response of hours to a positive productivity shock has also become an important point of difference between models that assume perfectly ﬂexible prices versus those that assume nominal price rigidity. We discuss New Keynesian models with monopolistic competition and prices that are ﬁxed in the short run in Chapter 5.

3.3. INITIAL CRITICISMS
In perhaps what is the most famous example of an RBC model, Kydland and Prescott [141] formulated a real business cycle model in which preferences are not separable over time with respect to leisure and there exists a time-tobuild feature in investment for new capital goods. The production function displays constant returns to scale with respect to hours worked and a composite capital good. The only exogenous shock to their model is a random technology shock which follows a stationary ﬁrst-order autoregressive process. They use the quadratic approximation procedure to obtain linear decision rules for a set of aggregate variables, and generate time series for the remaining series by drawing realizations of the innovation to the technology shock. They calculate a small set of moments associated with each series to match the model with the data. When calibrating their model, Kydland and Prescott choose the variance of the innovation to the technology shock to make the variability of the output series generated by their model equal to the variability of observed GNP. As McCallum [156] notes, this feature of their analysis makes it difﬁcult to judge whether “technology shocks are adequate to generate output, employment, etc. ﬂuctuations of the magnitude actually observed.” One of the most pervasive criticisms of the model is that there is no evidence of large, economy-wide disturbances that can play the role of the technology shocks posited by Kydland and Prescott. The only exception is oil price shocks, leading one to question whether the model was developed in reaction to the supply-side disturbances that characterized the 1970s. The RBC approach in general has difﬁculty in answering what some other examples of big shocks are. Summers [202] argues that “the vast majority of what Prescott labels technology shocks are in fact observable concomitants of labor hoarding and other behavior which Prescott does not allow for in his model.” This point was subsequently

Models of Business Cycles

47

elaborated on by Hall [110, 111], who argued that the procyclical movements in the Solow residual were, in fact, endogenous changes in efﬁciency arising from labor hoarding, increasing returns to scale, or price markups. Summers [202] also took issue with the parameters of the model that Kydland and Prescott [141] or Prescott [173] advocated in their calibration exercise. He argued that the econometric evidence presented by Eichenbaum, Hansen, and Singleton [83] on the intertemporal substitution of leisure hypothesis pointed to a share of time allocated to market activities that was more in the range of one-sixth, not one-third. He also noted that a real interest rate of 4% was inconsistent with the historical evidence of very low yields averaging around 1%. Singleton [192] raised the problem of inference and sampling uncertainty surrounding the various measures of ﬁt proposed in the RBC literature. He noted that the calibration approach did not provide a straightforward and transparent way of judging whether a given model constituted an improvement over another model. Eichenbaum [82] raised the issue of the model’s ﬁt based on the ratio of variance for GDP implied by the model and the data. Eichenbaum showed that the standard error of this ratio was heavily inﬂuenced by the parameters of the assumed stochastic process for the technology shock. The policy implications of the RBC model have also come under attack. According to the RBC model, business cycle ﬂuctuations are Pareto optimal and there is no role for the government to try to smooth or mitigate the ﬂuctuations. In fact, such policy efforts are inefﬁcient. Prescott [173] commented as follows:
Economic theory predicts that, given the nature of the shocks to technology and people’s willingness and ability to intertemporally and intratemporally substitute, the economy will display ﬂuctuations like those the U.S. economy displays. . . . Indeed, if the economy did not display the business cycle phenomena, there would be a puzzle.

Thus, according to this approach, cyclical ﬂuctuations are viewed as the natural response of the economy to changes that affect individuals’ consumption or production decisions. However, if prices or wages are ﬁxed or rigid or if there are other types of frictions arising from credit markets, then the policy implications of the RBC approach may not be valid and the economy may operate with high levels of unemployment and excess capacity over long periods. In this

48

Business Cycles: Fact, Fallacy and Fantasy

case, government policies may be required to move the economy towards a full employment level. In the chapters that follow, we will discuss the efﬁcacy of the RBC model in serving as a model of aggregate ﬂuctuations as well as questions that have lingered to this day about the adequacy of the calibration approach in matching the model to the data.

3.4. “PUZZLES”
Many of the initial criticisms which had been voiced by opponents of the RBC approach took on the character of “puzzles”. These puzzles have constituted the basis of much further research in the area. We can list some of these puzzles as follows: • The variability of hours and productivity: As seen in Table 3.1, one of the main problems with the standard RBC model is that it cannot capture the variation in the aggregate labor input (see also Kydland [140]). In the data, employment is strongly procyclical and almost as variable as output while real wages are weakly procyclical. In the standard model, a productivity shock shifts the marginal product of labor so that the observed variations in employment can only occur if the labor supply curve is relatively elastic. Yet, micro studies ﬁnd that wage elasticity of labor supply is quite low. In this case, the marginal productivity shock should lead to most of the adjustment in real wages and less in the quantity of labor. Furthermore, in the data, hours of work per worker adjusts very little over the cycle. About two-thirds of the variability in total hours worked comes from movements into and out of the labor force, and the rest is due to adjustment in the number of hours worked per employee. These ﬁndings create a puzzle for the standard RBC model. • The productivity puzzle: In the data, the correlation between productivity and hours worked is near zero or negative, while the correlation between productivity and output is positive and around 0.5.7 By contrast, the RBC model, which is driven entirely by productivity shocks, generates correlations that are large and positive in both cases. Another problem that
7 See, for example, Christiano and Eichenbaum [65], who measure hours worked and productivity based on both household and establishment-level surveys conducted by the US Department of Labor.

(4. Let πt denote the probability that a given agent is employed in period t so that the number of per capita hours worked is given by ˆ Ht = πt h. A Model with Indivisible Labor Supply
The indivisible labor. This economy is one in which individuals and a ﬁrm trade a contract that commits the household to work h0 hours with probability πt . • Reverse causality: The stylized facts of business cycles state that money.
3. labor’s share of income moves countercyclically. the individual gets paid regardless of whether he works or not. We now describe a version of the RBC model due to Rogerson [179] and Hansen [112] that allows for ﬁxed costs and nonconvexities in labor supply. especially M2. (4. this is inconsistent with the notion that TFP shocks are the driving force behind business cycle activity.4. where ˆ 0 < h < 1. ˆ Suppose that workers are constrained to work either zero or h hours. whereas in the RBC model labor’s share is ﬁxed. This is accomplished by assuming that individuals can work all the time or not at all.Models of Business Cycles
49
arises in matching the model and the data is that in the data. πt .31)
. However. lottery models studied by Gary Hansen [112] and Richard Rogerson [179] have been used to explain the stylized fact that aggregate hours vary more than productivity does. The conventional wisdom regarding the money-output correlation. summarized by Friedman and Schwartz [94] in their study of monetary history. A lottery then determines whether an individual actually works. appears to be a leading indicator of aggregate output.1. Since what is being traded is the contract. This framework also provides a way for reconciling large labor supply elasticities at the aggregate level with low labor supply elasticities at the individual level.30)
The main idea is that there are nonconvexities or ﬁxed costs that make varying the number of employed workers more efﬁcient than varying hours per worker. it is assumed that individuals choose the probability of working. To account for the nonconvexities introduced by the work/nonwork decision. is that the causality goes from money to output but with “long and variable lags”.

πt c1.t
(4.t πt = πt λt . 1).32)
max E [u(ct .50
Business Cycles: Fact. lt )] = πt u(c1. Then the expected utility of the representative consumer.t + (1 − πt )c0.
Notice that the social planner chooses the consumption allocations of each agent plus the probability of their working.t denote the consumption of an unemployed worker and c1. c1. Hence.t . c1. l ) = ln (c) + A ln (l ). the ﬁrst-order conditions with respect to (c0. Notice that the agent will consume ct whether or not he is working. taking into account the work versus nonwork decision. When agents do work.c1. expected utility (where the expectation is over whether or not you work) is ˆ ln (ct ) + πt A ln (1 − h) + (1 − πt )A ln (1).33) (4.35)
. (4. Let λt denote the Lagrange multiplier on the feasibility constraint for consumption.t = ct so that the agent consumes the same amount whether or not he is working. Hence. In this model. c0. ex ante all individuals are alike.34)
It follows that c0.t .t .c0. all individuals have the same consumption but the unemployed are better off. Fallacy and Fantasy
Let c0.t = c1. they ˆ must supply h hours of work so that there is no choice over hours of work directly. Omitting the work/leisure decision for the moment. 1 − h) + (1 − πt )u(c0. the unemployed worker enjoys higher utility since working causes disutility.t denote the consumption of an employed agent.t. This is a feature that is counterfactual to the working of actual labor markets. The social planner solves the problem
πt . Assume that the individual utility function has the form u(c. but ex post they differ because some work while others enjoy leisure. With complete insurance and identical preferences that are separable with respect to consumption and leisure.t (4. is given by ˆ E [u(ct .t . lt )] s.t = ct .t ) are 1 − πt = (1 − πt )λt .

and Heckman [48] for further discussion regarding the reconciliation of micro evidence with dynamic general equilibrium models.4.8
3. The purpose of the framework that Hansen [112] and Rogerson [179] consider is to generate the stylized fact with respect to the relative variability of hours versus productivity.52
Business Cycles: Fact. δ. A.
.
(4. β. Hansen argues that the model with indivisibilities can generate a variability of hours relative to productivity around 2.40)
Equations (4.41)
where the term in square brackets shows the rate of return to investing in the aggregate production technology. Fallacy and Fantasy
Notice that (4.2.39) yield the intertemporal Euler equation as 1=β ct ct+1
θ−1 1−θ [exp (zt+1 )θkt+1 Ht+1 + (1 − δ)] .
(4.40). The Productivity Puzzle
Several resolutions of the productivity puzzle have been suggested. These typically specify an extra margin regarding individuals’ choices that helps break the close link between hours and productivity implied by the standard model.7 compared to the model without indivisibilities which implies a value near unity. we can use the resource constraint to solve for ct and then substitute for ct into the intertemporal Euler equation to obtain a nonlinear stochastic in kt+1 with forcing process {zt }.37) and (4. and it is not intended to incorporate the microeconomic foundations of the labor market.38) can be used to solve for Ht as Ht = Bct exp (zt )(1 − θ) ˆ A ln (1 − h)ct ˆ exp (zt )(1 − θ)h
−1/θ
kt
−1/θ
= −
kt . With Ht determined in (4. and the stochastic process for the technology shock using the approach in Kydland and Prescott [141]. Hansen [112] calibrates this model by specifying values for the unknown parameters θ. Hansen.
8 See Browning.

0 < θ < 1. namely. gt p
(4. (4. ¯ ln (N − nt ) ¯ (4.
(4.43)
is public consumption. Let N denote the time endowment of the representative household per period. Per capita output is produced according to the Cobb–Douglas production function yt = (zt nt )1−θ ktθ . They consider a prototypical RBC model with a labor/leisure choice.
p where ct is private consumption. and α is a parameter that governs the impact of government consumption on the marginal utility of private consumption.45)
where zt is a technology shock which evolves according to the process zt = zt−1 exp (λt ).44) V ( N − nt ) = ¯ N − nt
for all t.46)
. one which is based on a time-separable logarithmic speciﬁcation and a second which incorporates the Hansen indivisible labor assumption. They consider two different speciﬁcations for the labor/leisure choice. and utilize the solution of the social planner’s problem to derive the ¯ competitive equilibrium allocations. They generate the negative correlation between hours worked and real wages or productivity by introducing government consumption shocks. The social planner ranks alternative consumption/leisure streams according to
∞
E0
t=0
¯ βt [ln (ct ) + γV (N − nt )] .42)
¯ where ct denotes consumption and N −nt denotes leisure of the representative household. (4. the standard model cannot deliver the strong procyclical response of hours without procyclical behavior in productivity.Models of Business Cycles
53
Government Consumption Shocks
Christiano and Eichenbaum [65] note that as long as there is a single shock that drives the behavior of both hours and productivity. Consumption services ct are related to private and public consumption as follows: ct = ct + αgt .

The aggregate resource constraint stipulates that consumption plus investment cannot exceed output in each period: ct + gt + kt+1 − (1 − δ)kt ≤ yt . λt is an independent and identically distributed process with mean λ and standard deviation σλ . u3 < 0. |ρ| < 1. consider a decision-maker who has preferences
∞
βt u(cmt .
t=0
where 0 < β < 1. hmt is labor time spent in market work. hours of work can increase along a downward-sloping labor demand curve. in the absence of technology shocks. Following Benhabib. Rogerson. hours increase and average productivity declines in response to a positive government consumption shock. If leisure is a normal good. cmt is the consumption of a market good.
Home Production
Another way of improving the model’s ability to match the data is to introduce home production. ct = ct /zt . the solution for the model can be more fruitfully expressed in terms of the transformed variables ¯ ¯ ¯ ¯ kt+1 = kt+1 /zt . g (4. In this expression. hht ). and u4 < 0. The key assumption is that private and government consumption are not perfect substitutes. cht is consumption of the home-produced good. yt = yt /zt . and gt = gt /zt . The total amount of time available to the household is normalized as unity.
. and Wright [39].47)
Since the technology shock follows a logarithmic random walk.48) where ln (¯ ) is the mean of ln (¯t ). Hence.54
Business Cycles: Fact. cht . and hht is labor time spent in home work.
p
(4. Assume that u1 > 0. an increase in government consumption leads to a negative wealth effect for consumers through the economy-wide resource constraint. and leisure is deﬁned as time not spent working in the market or at home: lt = 1 − hmt − hht . The speciﬁcation of the model is completed by assuming a stochastic law of motion for gt as ¯ g g ln (¯t ) = (1 − ρ) ln (¯ ) + ρ ln (¯t−1 ) + µt . hmt . u2 > 0. Fallacy and Fantasy
In this equation. and µt is the innovation to ln (¯t ) g g g with standard deviation σµ . government consumption shocks lead to shifts in the labor supply curve so that. In this model.

A rise in market productivity may induce households to substitute away from home production towards market production. one criticism of the home production theory is that it suggests that all movements out of the labor force (toward home production) are voluntary. This gives us another margin to substitute market labor and improves the model’s predictions. the labor supply curve also shifts in response to a good productivity shock. if there are costs to hiring or laying workers off. However. respectively.Models of Business Cycles
55
At each date. In the model. This says that the effective labor input can be altered even though the total number of workers is ﬁxed. the household can purchase market goods cmt . home production is used to produce a nontradeable consumption good. Letting wt denote the wage rate and δm and δh denote the depreciation rates on market and home capital. kht . but only after longer periods of time.t+1 + kh.
Labor Hoarding
A third way to resolve the productivity puzzle is through a labor hoarding argument. Household capital goods are used in home production.t+1 ≤ wt hmt + rt kmt + (1 − δm )kmt + (1 − δh )kht . the household’s budget constraint is cmt + km. Unlike the standard model. where zht is a shock to home production and g is increasing and concave in labor and capital. market capital goods kmt . Alternative measures put home production to be in the range of 20–50% of GDP. Hence. labor effort is likely to be adjusted ﬁrst in response to a productivity shock. Eventually more workers may be hired or ﬁred. but market capital goods are rented to ﬁrms at the competitive rental rate rt . thereby leading to greater variability in labor. However. zht ). then ﬁrms may retain workers even though they are not exerting much effort. Home goods are produced according to the home production function: cht = g (hht . and household capital goods kht . The ﬁrm may not alter its work force every time there is a productivity shock (which would occur through a shift in the marginal product of labor curve).
.

let f denote a ﬁxed shift length. In the absence of variations in work effort. the work effort of each individual.
3. capital Kt . If labor hoarding is included in the model. Yt = At Ktα [fNt Wt ]1−α . This can lead to variation in the Solow residual that is not related to changes in productivity or technology. Speciﬁcally. Nt .3. and Rebelo [52]. To brieﬂy describe their framework. then the productivity/hours correlation is reduced and more closely matches the data.4. Notice that the conventionally measured Solow residual St is related to true technology shock At as ln (St ) = ln (At ) + α ln (Kt ) + (1 − α)[ln (f ) + ln (Nt ) + ln (Wt )] − α ln (Kt ) − (1 − α)[ln (f ) + ln (Nt )] = ln (At ) + (1 − α) ln (Wt ).
In the standard model. the production function can be written as Yt = St Ktα [Ht ]1−α . the total number of workers. Reverse Causality
One approach to dealing with this criticism is to introduce money and banking into the standard RBC model following King and Plosser [129]. Eichenbaum. and a labor input that reﬂects variations in work effort following Burnside. Decisions to alter effort levels will be the outcome of a maximizing decision. where Ht denotes the total hours worked. These authors observe that transactions services (as provided by money and the banking sector) can be viewed as an intermediate input that reduces the cost of producing output and. Fallacy and Fantasy
To illustrate the role of labor hoarding. so that the movements in the Solow residual are not entirely exogenous. consider a production function for aggregate output Yt which depends on an exogenous technology shock At . suppose
. The comments made regarding ﬂuctuations in the effort level of labor also apply to capital. Ht = fNt . and Wt . 0 < α < 1. The measured capital in the Solow residual does not take into account optimal ﬂuctuations in capital utilization rates. hence. Thus.56
Business Cycles: Fact. can be treated as a direct input into the aggregate production function.

The ﬁnancial industry is assumed to provide accounting services that facilitate the exchange of goods by reducing the amount of time that would be devoted to market transactions. where wt is the real wage and ρt is the rental price of transactions services. nft is the amount of labor services. The production of the intermediate good is given by dt = g (ndt . (4.51)
Households are assumed to own the capital stock and to make investment decisions it subject to the resource constraint ct + it ≤ yt + (1 − δ)kt .Models of Business Cycles
57
the ﬁnal good is produced according to the production technology yt = f (kft . nft . dft )φt .52)
where τ < 0. Finally. In this expression. (4. hours allocated to ∗ producing transactions services is nτt = τ ∗ (h(ρt /wt ))(ct + it ). ﬁrms rent labor.
0 < β < 1. The household chooses an amount of transactions services dht so as to minimize the total transactions costs. capital.
(4.50)
where ndt and kdt denote the amounts of labor and capital allocated to the ﬁnancial sector. we
. φt is a shock to production of the ﬁnal good at time t. This implies a demand for transactions services that can be obtained from the ﬁrst-order condition ρt = wt τ (dht /(ct + it )) (4. where 0 < δ < 1 is the depreciation rate on capital. and λt captures technological innovations to the ﬁnancial services industry. Households maximize the expected discounted value of utility from consumption ct and leisure lt as
∞
E0
t=0
βt U (ct . By contrast. and dft is the amount of transactions services used in the ﬁnal goods industry. where h = (τ )−1 . respectively. (4. and transactions services to maximize proﬁts on a period-by-period basis. Households are also assumed to combine time and transactions services to accomplish consumption and investment purchases. Likewise.53)
∗ as dht = h(ρt /wt )(ct + it ). The time required for this activity is nτt = τ(dht /(ct + it ))(ct + it ). wt nτt + ρt dht . τ < 0. lt ) . kdt )λt .49)
where kft is the amount of capital.

Speciﬁcally. Suppose that a shock to the production of ﬁnal goods or. commercial credit. if interest rates are rising and ﬁrms wish to invest more in anticipation of higher expected proﬁts. a positive shock to productivity occurs. investment demand rises and output will also increase in response to the additional hours worked. Then. Consistent with the RBC view. there will be an increase in hours of work. This framework can be used to rationalize the observations regarding money and output. the central bank may expand the money supply to keep interest rates from rising too rapidly.
3.
. Ahmed and Murthy [3] provide a test of this hypothesis using a small open economy such as Canada to evaluate the impact of exogenously given terms of trade and real interest rates versus domestic aggregate demand and supply disturbances. they ﬁnd that an important source of the money-output correlation is output shocks affecting inside money in the short run. equivalently. The increase in output will also stimulate the demand for transactions services by households and ﬁrms.9 During an expansion.5. THE SOURCE OF THE SHOCKS
The RBC model has come under much criticism for assuming that technology or productivity shocks are the sole driving force of cyclical ﬂuctuations. If the substitution effect of an increase in the marginal product of labor outweighs the wealth effect of the productivity shock. responds to the positive productivity shock. As a consequence. Their results are derived from a structural vector autoregression (VAR) with long-run restrictions to identify the alternative structural shocks. is more closely related to output than outside money. One
9 See Sims [190].58
Business Cycles: Fact. the causality runs from the productivity shock to money even though the increase in money occurs before the higher productivity shock. Fallacy and Fantasy
require that the household’s time allocated to the different activities sums to 1. nt = nft + ndt + nτt . or a broad measure of money that includes commercial credit. Another possibility is that the central bank uses the accommodative monetary policy. Hence. As a result. inside money. or inside money. consumption and leisure of the representative consumer will rise but so will investment demand as consumers seek to spread the extra wealth over time. so ﬁrms will wish to ﬁnance a greater volume of goods in process.

46) between a de-trended measure of the relative price of capital and investment expenditures. In the previous section. the relative price of new equipment declined by 3% at an average annual rate while the equipment-GNP ratio increased substantially. Demers. Hercowitz.1. during the period 1950–1990. 11 In Greenwood. These authors consider the impact of distortionary taxes on real outcomes. they nevertheless show that political events can have non-negligible effects on real economic variables. Other papers that have employed the RBC framework with ﬁscal shocks include Braun [46] and McGrattan [158]. and Demers [10] examine the impact of political risk and regime changes due to changes in the party in power on real investment decisions under irreversibility. the role of investment-speciﬁc technological shocks is used to account for cyclical ﬂuctuations. and Krusell [106. This has been examined by Greenwood.
. we discussed the role of government or ﬁscal shocks in generating observed co-movements of the data.
0 < θ < 1.Models of Business Cycles
59
way of dealing with this criticism is to introduce alternative sources of shocks into RBC models. Although their analysis is not general equilibrium. thereby stimulating output growth. these authors motivate their analysis by noting that. Christiano and Eichenbaum [65] introduce government consumption shocks as a way of breaking the strong and positive relationship between hours worked and productivity implied by the basic RBC model. 107]. Investment-Speciﬁc Technological Shocks
An important source of shocks is the investment-speciﬁc technological change. [107]. War shocks can also be modeled as a source of cyclical ﬂuctuations (see Barro [27] and Ohanian [169]). Hercowitz. The notion behind this type of change is that production of capital goods becomes increasingly efﬁcient over time.11 In Greenwood et al. and Krusell [106].10
3. Following Gordon [104].
10 Altug. Furthermore. the authors use a growth accounting approach to show that investment-speciﬁc shocks can account for 60% of postwar US growth of output per man-hour. there was a negative correlation (−0.5. In this vein. Consider a standard RBC model with a representative consumer who derives utility from consumption and leisure as
∞
E0
t=0
βt [θ ln (ct ) + (1 − θ) ln (1 − lt )] .

0 < αe . κs > 0.56)
Thus. The production function is given by y = F (hke . Structures evolve according to the standard law of motion: ks = (1 − δs )ks + ii . the service ﬂow is denoted hke . (5. l . where ζt+1 and ηt+1 are innovations to productivity and investmentspeciﬁc technological change. (5.54)
The law of motion for equipment is given by ke = (1 − δe (h))ke + ie q. where ae = exp (η)φe (ke /q − κe ke /q)2 /(ke q). that follow ﬁrst-order Markov processes.55)
where the variable q shows the investment-speciﬁc technological change in equipment and δe (h) = b h. φe . respectively. 0 < δs < 1. κe > 0. αe + αs < 1.60
Business Cycles: Fact. equipment denoted ke and structures denoted ks . respectively. αs . Investment in both structures and equipment is also subject to convex costs of adjustment denoted as a = ae + as . The revenue raised by the government is
. The shocks to technology and to the efﬁciency of equipment capital constitute the drivers of the t+1 t+1 model. as = φs (ks − κs ks )2 . (5. ks . there is a government which levies taxes on labor and capital income at the rates τl and τk . z) = z(hke )αe ksαs l 1−αe −αs .58)
Finally. They evolve as zt+1 = γz exp (ζt+1 ) and qt+1 = γq exp (ηt+1 ). Equipment is utilized at the time-varying rate h. Fallacy and Fantasy
where ct denotes consumption and lt denotes labor supply. (5. ω ω > 1. investment in equipment is subject to variable rates of utilization and depreciation which are convex in the utilization rate. and γz and γq denote the average gross growth rates of the two processes. hence.
where z is a measure of total factor productivity. φs .57) (5. There is a production technology that depends on labor and the services from two types of capital goods.

ae . the depreciation rate on structures δs . and ks all have to grow at the same rate. denote this rate by g . this quantity is determined as the ratio of Gordon’s [104] price index of quality-adjusted equipment and the price index for consumption. or z q
g = γz ge = γz
1/(1−αe −αs ) αe /(1−αe −αs ) γq . and the tax rate on wage income τl . The resource constraint requires that consumption plus investment in the two types of capital equal output net of adjustment costs: c + ie + is = y − ae − as . Equipment. ie . as .59)
where re and rs denote the rental rate on equipment and structures. respectively. However.60)
In this analysis. the investment-speciﬁc technological shock plays a key role. The resource constraint and the accumulation equation for structures imply that y. c. respectively. (5. A higher value of q directly affects the quantity of equipment that will be available to produce output next period.12 A key parameter to be estimated from the data is γq . Hence. it also affects the utilization cost of existing equipment this period by lowering the replacement cost of old capital goods. In the balanced growth path equilibrium. and w denotes the real wage rate. investment-speciﬁc technological change increases the services from old equipment at the same time as it increases the quantity of equipment available for next period. The ﬁrst parameter is determined using data on the inverse of the relative price of investment goods following Gordon [104]. and hence increases its utilization.
. is . We note that the economy displays balanced growth as zt and qt grow at the rates γz and γq . The parameters that are determined using a priori information include the average growth rate of the investment-speciﬁc shock γq . This yields the government budget constraint: τ = τk (re hke + rs ks ) + τl wl . grows faster. we have that g = γ (γ g )αe g αs . (5.Models of Business Cycles
61
returned to the private sector through lump-sum transfers τ. however. the quantity of labor l and the utilization rate h are constant. Since the inverse of q denotes the relative price of equipment in terms of consumption goods. 1/(1−αe −αs ) (1−αs )/(1−αe −αs ) γq . at the rate ge = γq g . and it is set at
12 From the production function.

For this purpose. A high value of φ works to reduce investment and to increase the procyclicality of consumption. the symmetry condition φe = (g /ge )2 φs ≡ φ leaves only the parameter φ to be determined. Likewise.40. Given that equipment investment is only 7% of GNP and 18% of the value of output being derived from the use of equipment.40. This constitutes an upper value for the contribution of the shocks q. the impact of the investment-speciﬁc technological shock depends on the quantitative importance of the transmission mechanism. Unlike the standard technology shock in RBC models. They ﬁnd that under a high value of φ. where the parameters b and h cannot be identiﬁed separately. The restriction that adjustment costs are zero along the balanced growth path yields the values κe = κs = g . For a low value of φ. β = 0. This standard calibration exercise yields θ = 0. the fraction of output directly affected by the shock will typically be small.
. only the q shock is assumed to be operative. By contrast.62
Business Cycles: Fact.032.18. τk = 0. which raises utilization of equipment today. According to the ﬁrst. αe = 0. The model is simulated to understand the contribution of the investmentspeciﬁc technological shocks in generating cyclical ﬂuctuations. This provides a lower bound on the contribution of the q shocks because a positive shock to q reduces consumption at the same time as it increases investment. a low value of the adjustment cost parameter φ is chosen to make the volatility of investment in the model equal to that in the data. this ﬁgure is 32%. g = 1.056 and τl = 0. Furthermore. the parameter φ is set to equalize the consumption/output correlation implied by the model to that in the data. Fallacy and Fantasy
1. Greenwood et al. Hence. ω = 1. This leads to increased employment and output. and bh ω = 0.59. αs = 0. there is a decline in the equipment’s replacement value. Simultaneously. δs = 0. The second set of parameters is calculated using information on the long-run behavior of a set of the model’s variables. they are not the main factor. They conclude that though investment-speciﬁc shocks contribute to business cycle ﬂuctuations.97. Hence.12. which tends to raise the stock of equipment next period. the investment-speciﬁc shock affects uses of factors such as labor and capital through changed investment opportunities. the investment-speciﬁc technological shocks explain around 28% of business cycle ﬂuctuations as captured by the standard deviation of output. [107] calibrate the adjustment cost parameter under two alternative assumptions. A positive shock to this variable works by increasing the rate of return to equipment investment.0124.53.20.

ω1
where 0 < δ(ut ) < 1. capital utilization is the channel through which energy shocks enter the model. We brieﬂy describe her model before concluding this chapter. By contrast. kt is the stock of capital. Finn [92] argues that it is possible to rationalize the role of energy shocks in models with perfect competition. δ(ut ) = ω0 utω1 .
. ν1 > 1. the depreciation of physical capital depends on its utilization rate so that kt+1 = (1 − δ(ut ))kt + it . and ut is the rate of utilization of capital. lt ) = t 1−σ
1−σ
−1
. Rotemberg and Woodford [183] provide a rationale for the role of energy price shocks in a model with imperfect competition.
The production function is given by yt = (zt lt )θ (kt ut )1−θ . The new feature in Finn’s analysis is that capital utilization requires energy: et ν0 utν1 = . kt ν1 ν0 > 0. lt ) . Barsky and Kilian [28] analyze the relationship between oil shocks and changes in economic activity. As in the standard RBC framework. ω0 > 0. there is a representative consumer with preferences:
∞
E0
t=0
βt u(ct .2.
with c α (1 − lt )1−α u(ct .
0 < β < 1.
0 < α < 1. the source of large supply-side shocks for the RBC agenda has been sought in oil shocks. 0 < θ < 1.5.
where zt is an exogenous technology shock. σ > 0. As in the previous model.Models of Business Cycles
63
3. Energy Shocks
From the beginning. In her model. and ω1 > 1.

ut . Fallacy and Fantasy
We can solve this equation for the utilization rate ut and substitute it into the production function to obtain yt = (zt lt )
θ 1− ν1 ν1 kt 1 et 1
ν1 ν0
1 ν1
1−θ
. This shows the direct effect of energy.64
Business Cycles: Fact. Finn [92] describes qualitatively and quantitatively the impact of increases in the price of energy. and lt continue into the future and reduce the future marginal productivity of capital. If the price shock is persistent. a positive energy price shock operates in a similar way as a negative technology shock. This tends to reduce the investment it and the future capital stock kt . capital. and energy. Finn [92] shows that the quantitative response of value-added and real wages to a shock in energy prices implied by the model is in line with that in the data. The indirect effect of the increase in the price of energy on capital’s future marginal energy cost also tends to reduce investment and the future capital stock. and hence the real wage wt and work effort lt . the economy-wide resource constraint is given by ct + it + pt et ≤ yt . The decrease in et further reduces the marginal product of labor. then the decrease in the values of et . First. Thus. the production of output depends on labor. an increase in pt reduces et and ut through the production function. Hence.
.
Thus. Finally. there are two channels for the transmission of energy shocks in the model. where pt is the price of energy. but there is also an indirect effect which works through the effect of the utilization rate on the depreciation of capital.

Unlike the closed-economy model. Kehoe. and Kydland [25] consider data on 12 developed countries over a sample period dating from 1960 to 1990. Backus. open-economy business cycle models allow for the role of technology processes in different countries in generating cyclical ﬂuctuations. we review some of the facts of international business cycles and discuss models that have been used to rationalize the ﬁndings. [26] study
65
.
4. Backus et al. They consider a two-country RBC model that has the features of the original Kydland–Prescott model and that assumes a single homogeneous good and complete contingent claims for allocating risk. One of the aims of the RBC approach has been to determine whether two-country versions of the basic model can account for both the co-movements studied in closed-economy models as well as international co-movements. including correlations of macroeconomic aggregates across countries and movements in the balance of trade. They can be used to model the impact of the ability to trade in goods and to borrow and lend internationally.1. FACTS
One of the important areas of research has been to derive the so-called stylized facts of international business cycles. nor can it be used to discuss notions of international risk sharing. The closed-economy model cannot be used to understand the propagation of shocks across countries or regions.Chapter 4
International Business Cycles
The international business cycle literature provides an extension to the original real business cycle (RBC) approach by allowing for open-economy considerations. They study the properties of a baseline model against the actual features in the data. In this chapter.

The positive t t off-diagonal entries allow for spillover effects of the shock to productivity in one country to have a positive effect on the productivity of the other country. Thus. the exercise in this literature is to replicate the correlation properties among the different series given the variability and correlation properties of the shocks. Switzerland. Fallacy and Fantasy
the properties of the same model for ten developed countries plus a European aggregate developed by the OECD between 1970 and 1990. By contrast. Italy. In this regard. Japan. The price anomaly: The volatility of the terms of trade. The individual countries they consider are Australia. the cross-country correlations of output are greater than those of productivity. Canada. The large autoregressive coefﬁcients displayed on the diagonal capture the persistence in observed productivity.
•
. ∗ denote the respective innovations to these t shocks. Austria. 26] estimate a bivariate AR(1) process for the domestic and foreign shocks as zt zt∗ =A zt−1 + ∗ zt−1
t ∗ t
.906 0. ∗ ) = 0. zt∗ denote the technology shocks to the domestic and foreign countries.906
with Std ( t ) = Std ( ∗ ) = 0.
where zt .00852 and Corr( t . [25. and the United States. and the cross-correlations of productivity are greater than those of consumption. the United Kingdom.088 0. is much higher in the data than it is in the theoretical model.258. Based on several estimated speciﬁcations for the US and the European aggregate. The ﬁndings can be summarized under two headings: • The quantity anomaly: In the data. Germany. France. The data are ﬁltered using the Hodrick–Prescott ﬁlter.66
Business Cycles: Fact. measured as the Solow residual. and t . Among the important features of their analysis is the estimation of the correlation properties of international productivity shocks using data on estimated Solow residuals.088 . Backus et al. these authors use a symmetric speciﬁcation with A= 0. deﬁned as the relative price of imports to exports. 0. the theoretical model implies the reverse ranking. respectively.

This leads to a strong investment response in the US. They estimate the moments of interest using the generalized method of moments (GMM) estimation. the US. implying that the real exchange rate is also less volatile in the model than it is in the data. and Zimmermann [14] extend the Backus et al. investment. The international RBC model produces these results because it implies that the incentive is to use inputs where they are most productive.International Business Cycles
67
Ambler. what is typically observed in the data are positive comovements in these series across countries. which is accompanied by output and employment increases. but the variability of investment and the trade balance continue to remain
. investment and the trade balance are more volatile in the model than they are in the data. Cardia. [25] also examine a variety of perturbations of their original setup. They also ﬁnd that the cross-country correlations of output. Backus et al. By contrast. they are negative. whereas in the model. This fact leads to negative cross-correlations between output. Since investment responds strongly to a positive productivity shock. which allows them to obtain standard errors. The price anomaly arises because the real exchange rate in the baseline international RBC model is closely related to the ratio of consumption across the two countries. which leads to a trade deﬁcit in the domestic country. This feature of the model leads to the cross-country correlations of consumption being much higher in the model than in the data. First. [25. [14] conﬁrm the ﬁndings of Backus et al. though with lower magnitudes. The ﬁndings of Ambler et al. they consider asymmetric spillovers between the US and the European aggregate in which the response of US productivity to shocks to European productivity is smaller than the response of European productivity to shocks to US productivity. and employment are positive and generally high in the data. we thus observe a ﬂow of factors from other countries or regions to the US. This changes the outputinvestment correlation from positive to negative in the domestic country. In a model with perfect capital mobility. Furthermore. 26]. and employment in the theoretical model. this ratio displays little volatility. investment. the increase in investment plus consumption is greater than the increase in output. say. In other words. suppose a positive technology shock occurs in one country. With perfect risk sharing. In their original analysis. [26] sample to 20 industrialized countries over the period 1960–2004. The two-country frictionless international business cycle model also implies that consumers in different countries can share risk perfectly.

Backus et al. Fallacy and Fantasy
counterfactually high.258 to 0. Cole [71] argued that it is important to examine how alternative ﬁnancial arrangements interact with the preferences and the production possibilities set to determine the behavior of the real variables. they consider large spillovers across countries by changing the off-diagonal elements of the A matrix from 0. They ﬁnd that the results are very similar to the case with a small transport cost. it is important to understand the role of alternative ﬁnancial arrangements in generating the cross-country correlations. as does the positive correlation of consumption across countries. The standard international RBC model studied by Backus et al. However.5. Second. Cole [71] and Cole and Obstfeld [72] examine a two-country real version of the Lucas asset pricing model for this purpose (see Lucas [152]).088 to 0. Next. The ﬁrst of these involves introducing a small trading friction in the form of a transport cost.2 and the correlation of the innovations from 0. With this change. Hence. the foreign agent increases consumption and reduces investment in anticipation of future income increases. Agents from both
. but has little effect on the cross-country correlations of consumption and output. but stems from the permanent income hypothesis: when domestic and foreign productivity shocks are correlated.2. In this case. [25. elimination of trade in state-contingent claims does not help to lower the cross-country correlations of consumption. they ﬁnd that investment and the trade balance become much less volatile. They also examine various sorts of trading frictions. Surprisingly. they eliminate all trade in goods and assets by considering an autarky situation. [25] argue that this result is not due to international risk sharing considerations. and the crosscountry correlation of output goes from being negative to positive.68
Business Cycles: Fact. it is the correlation between technology shocks that leads to any connection between countries. THE ROLE OF INTERNATIONAL RISK SHARING
Before we continue with other extensions that have been proposed in the international RBC literature.
4. a rise in productivity in the domestic country signals a rise in productivity in the foreign country through the spillover effects. the cross-country correlation of consumption increases further. 26] considers the case with complete contingent claims and perfect risk sharing across different countries. They ﬁnd that this friction has the effect of reducing the volatility of investment and the trade balance.

t 1 c2.t = (1 − δ)y1. It is straightforward to demonstrate
.t ) =
α 1−α (c1.10)
where δ= 1+ φ2 φ1
σ
1 and σ = ρ . These expressions show that consumers in each country consume a constant fraction of output in each period. which also causes the output of country 2 to increase through positive spillover effects. the Pareto optimal consumption allocations satisfy
1 c1.International Business Cycles
71
Hence.6)
Assuming that preferences are of Cobb–Douglas variety.t )1−ρ
1−ρ
.2.t = δy1.t 2 c1. This implication of the model has been shown to lead to the counterfactually high cross-country correlations of consumption implied by the standard international RBC model.t c2.9) (2.t 2 c2. c2.7) (2.
(2. In what follows.
(2.2.
4. Also note that a shock to the output of country 1. we ﬁnd that the ratio of marginal utilities across agents in different countries is equalized for all states and goods. This is an implication of perfect risk sharing that arises in a complete contingent claims equilibrium. we will examine the role of alternative ﬁnancial arrangements that can be used to rationalize the observations.t . It is convenient to assume that preferences are of the Cobb–Douglas variety: U (c1. Complete Contingent Claims
Consider ﬁrst the behavior of allocations in a competitive equilibrium in which agents can trade claims that pay off contingent on all possible realizations of the income processes in each country.t = (1 − δ)y2.t = δy2.t . will cause consumption in both countries to increase.8) (2.

t
1 1 q(st+1 . it is without loss of generality to consider only one-period contingent claims.t
q(st+1 .t .t
+
st+1 ∈S 2 z1.t
p(st )
2 + c2.t = c1. see Altug and Labadie [6]. Fallacy and Fantasy
that there exists a contingent claims equilibrium for this economy.1 Agents can also trade in the goods of the other country. st ) is the price of a contingent claim that pays off in each possible j state next period conditional on the state today and zi (st+1 ) are the holdings of country i’s assets by consumers from country j for i.72
Business Cycles: Fact. The current account for each country.t + p(st )c2. these equations indicate that the output produced in country 1 plus the receipts (or payments) on assets denominated in the goods of countries 1 and 2 must be equal to consumption of domestic goods plus imports of foreign goods plus purchases (sales) of contingent assets denominated in the goods of countries 1 and 2. denoted as CAj .t
(2.t − c1.t + z1.t −
2
2 c1. 2. is given by
1 1 CA1 = y1. 7.12)
where q(st+1 . Given the Markov nature of uncertainty.t . The sequential budget constraints for residents of country j for j = 1. Ch.
1 For further discussion. st ) z1 (st+1 ) + p(st )z2 (st+1 ) 2 c1.t + +
p(st )
2 + z2. j = 1. 2 are given by
1 1 1 1 y1. st )
st+1 ∈S
2 z1 (st+1 ) 2 + z2 (st+1 ) .t + p(st )c2.t + p(st )z2. and that the competitive equilibrium allocations are Pareto optimal. Let p(st ) denote the relative price of good y2 in terms of the numéraire good y1 . Consider ﬁrst the case when agents can trade one-period contingent claims that pay off in each state next period on the consumption good of each country. p(st )
(2.
.
CA = y2.11)
y2. Thus.t =
p(st )
2 + c2.

We can interpret the budget constraints for each country j by stating that the sum of the current account plus the capital account must be zero: CAj + KAj = 0, j = 1, 2.

In an economy with trade in international assets, a current account deﬁcit (CA < 0) must be balanced with a capital account surplus (KA > 0). This implies that a country which consumes more than it produces must be a net borrower or, equivalently, it must sell more assets to the rest of the world than it purchases from the rest of the world. The market-clearing conditions are given by
1 2 ci,t + ci,t = yi,t , 1 2 zi,t + zi,t = 0,

In this equilibrium, we ﬁnd that consumers in each country equate their marginal rates of substitution for each good across all possible current states. If preferences satisfy the Cobb–Douglas assumption, then consumers in each country consume a constant fraction of current output as speciﬁed in equations (2.7)–(2.10). This follows from the fact that the competitive equilibrium is Pareto optimal. Hence, p(st ) =
i (1 − α)c1,t i αc2,t

=

(1 − α)y1,t . αy2,t

In this case, the contingent claims contracts are considered to be consistent with the results on consumption for each country. Given the solution for

International Business Cycles

75

the relative price p(st ) and the consumption allocations given in equations (2.7)–(2.10), notice that the solution
i i z1 (st+1 ) = z2 (st+1 ) = 0, i i z1,t = z2,t = 0

with δ = α satisﬁes the consumers’ budget constraints and the market-clearing conditions. But this is a contingent claims equilibrium in which no assets are traded! In such an equilibrium, it is still possible to price the contingent claims using the expression in (2.14). In the next section, we will demonstrate explicitly that the Pareto optimal allocations can indeed be attained in a noasset-trading equilibrium. Since the price of any ﬁnancial asset can be obtained as a function of the contingent claims prices, it follows that any ﬁnancial asset can also be priced in such an equilibrium.

4.2.3. No Asset Trading
Now suppose that there is trade in goods but no trade in international assets. In this case, we are forcing the current account to equal zero in each period. Country 1 and 2’s budget constraints can be expressed, respectively, as
1 1 c1,t + pt c2,t = y1,t , 2 c1,t 2 + c2,t = y2,t .

(2.15) (2.16)

pt

Once again, this is a static problem because there are no assets for intertemporal j consumption smoothing and the endowment is nonstorable. Let µt denote the Lagrange multiplier on the budget constraint of country j. If we assume the Cobb–Douglas functional form for preferences, then the consumption allocations are
1 c1,t = αy1,t , 1 c2,t =

(1 − α)y1,t , pt

2 c1,t = αpt y2,t , 2 c2,t = (1 − α)y2,t .

αy2.t
This price can be substituted into the consumption functions above to show that
1 c2. A large and positive shock in the amount of good y1.t .
ρ −1
1−α α
and φ2 = 1 − φ1 . 1 2 c2.t = y1.
Notice that the no-asset-trading allocation is identical to the central planning allocation if α=δ= 1+ or φ = 1+
1
φ2 φ1
σ
. This exercise illustrates several points: • The absence of international capital mobility does not necessarily imply that the allocation is not Pareto optimal. Efﬁcient risk sharing can occur despite the lack of ﬁnancial assets and insurance.t = y2.t = αy2.t is positively transmitted to the residents of country 2 by the increase in demand (and hence the relative price) for good y2.t + c2.17) (2. Fallacy and Fantasy
Equilibrium in the two goods markets requires that
1 2 c1. efﬁcient risk sharing occurs through changes in the relative price of goods.t .t .t = (1 − α)y1.
•
. 2 c1. A large negative shock in the amount of a good is similarly transmitted across borders.t .76
Business Cycles: Fact.t . The international ratio of marginal utilities across countries is identical across goods and states.t .18)
Substitute the consumption functions into the market-clearing conditions and solve for the relative price to show that pt = (1 − α)y1.
(2. despite the absence of trade in ﬁnancial assets.t + c1. Hence.

t + wtb . by
1 1 1 p1. αw y2.t .t c2.t = p2.t = αy1. Let α1 .t c1. As Cole and Obstfeld [72] point out.t in units of wt .t + pt c2. The total consumption of countries 1 and 2 is
1 1 c1.
4.2.t α2 [wt1 + wt2 ] .t .t = (1 − α)y1. the positive transmission of shocks occurs because countries specialize in the production of goods.t in terms of wt and p2. despite the absence of ﬁnancial capital mobility.
The equilibrium relative prices satisfy p1. so that p1.t denotes the relative price of good y2.t + p2.
Notice that correlation of the total value of consumption of country 1 and country 2 is positive and equal to one. 2 2 c1.t = α1 [wt1 + wt2 ] . αw denote the expenditure shares under the assumption of Cobb–Douglas preferences and let w be the numéraire good.t + cw.t c1. and can show that real interest rates and real asset returns will be equal for the two countries.t + αy2.t denotes the relative price of good y1. under the assumption that the current account is zero.t y1. w] denote the realization of good w in country j. Call this third good w. αw y1. 2 2 2 p1.t + cw. Assume that both countries receive an exogenous and stochastic endowment of good w and let w j : S → W = ¯ [w. respectively.t c2. α2 .t + wta . Nonspecialization in Endowments
To illustrate the impact of nonspecialization.t
. Cole and Obstfeld [72] introduce a third good.t + (1 − α)y2.t + p2.t ≤ p2.t ≤ p1.4.t y2.International Business Cycles
77
•
•
Notice that we can price ﬁnancial assets in the model. Agents in countries 1 and 2 have budget constraints given.t + pt c2.

t t share of the endowment of wt . ( w 1 +w 2 ) and ( w 1 +w 2 ). Call this good n and assume that nj : S → N = [n. This helps to clearly distinguish between country-speciﬁc shocks. Hence. Notice that the relative price of wt may not adjust much if wt1 and wt2 are negatively correlated but the sum wt1 + wt2 ﬂuctuates very little. the country must export more of the good in which it specializes in production to ﬁnance the import of good wt .t . in the absence of trade in ﬁnancial assets. Fallacy and Fantasy
The consumption of good 1 by agents in countries 1 and 2. Let α1 . by deﬁnition. The intuition is that.t are. a condition for Pareto optimality. the country with a negative shock to the endowment of wt would like to run a current account deﬁcit by importing wt and borrowing against future endowment. under the assumption of Cobb–Douglas preferences. Since the current account must always be balanced. Nontraded Goods
Suppose now that the third good is a nontraded good. Shocks to y1. If these shocks are not perfectly correlated. are sector-speciﬁc shocks.t = αw 2 c1. there are gains to asset trading that improve risk sharing and allow diversiﬁcation.2. The shocks to wt1 and wt2 must be perfectly correlated for the allocation with no trade in ﬁnancial assets to be Pareto optimal. α2 . country-speciﬁc shocks.78
Business Cycles: Fact.t = αw
wt1 wt1 + wt2 wt2 wt1 + wt2
+ α1 y1. is constant as the total t t t t
w1
w2
4. wt2 . whereas shocks to wt1 . wt2 are not perfectly correlated.t . n] denotes the realization of good n in ¯ country j.
Similar expressions can be derived for the consumption of goods y2 and w. Under the
. and industry-speciﬁc shocks. then it is beneﬁcial to trade equity shares or other forms of ﬁnancial assets.5.t or y2. αn denote the expenditure shares under the assumption of Cobb–Douglas preferences and let y1 be the numéraire good. + α2 y1. which affect all sectors within a country. The ratio of marginal utilities of both countries for each good will be equal to a constant across all states. can be shown to satisfy
1 c1. when there are sector-speciﬁc shocks. where wt1 . only if the wt varies with st .

An agent in country 1 holds equity shares that are claims to the endowment stream for good y1 (the domestic good) and claims to y2 (the foreign good).t . 2 denote the shares of good i held by an agent in country j at the beginning of period t. If this sector constitutes a large fraction of consumption.t for j = 1.International Business Cycles
79
assumption of balanced trade and Cobb–Douglas preferences. nt are perfectly correlated.t
α1 y1.t [y1.t .t + q1.
j j j j j j j
(2. An agent’s budget constraint is z1. 2 and i = 1.t + p2.t+1 + q2.t . n1
4. 1 − αn
1 2 Each country consumes its endowment of the nontraded good. We now discuss the impact of asset trading and relate it to the model without asset trade discussed earlier. the correlation of national consumption levels may be close to zero. nt .t y2. 1 − αn α1 = y2. nt . Thus. j Let zi.t+1 .t 2 c1.6. Notice that the correlation of consumption across countries will now depend on the proportion of a country’s consumption that is nontradeable.t [p2.2. The ratio of marginal utility across countries for a traded good will now depend 1 2 on the ratio nt2 .t 2 c2.t = 1 c2.t c2.t z2. then the resulting t allocations will not be Pareto optimal. then even if consumption of traded goods is perfectly correlated. There are gains from international risk sharing through asset trade. nontraded goods provide one vehicle for reducing the large cross-country correlations implied by the baseline model.t + q2.t z1.t .19)
. Unless nt . the demands for the goods satisfy
1 c1. Trade in Equity Shares
We now introduce trade in ﬁnancial assets.t ] + z2.t ] ≥ c1.t + q1. 1 − αn α2 = y2. 1 − αn α2 = y1.

U1 (c1. This simple example. In such a world. when combined with our discussion of the equilibrium with no trade in ﬁnancial assets. 2 so that φj = 1/2 and δ = 1/2. then portfolio diversiﬁcation may be achieved by specializing in the holding of equity shares of one country only. all equity shares are held and the endowment of each good is completely consumed. then portfolio diversiﬁcation may require that an agent hold equity shares for w 1 and w 2 if the endowment shocks for w are not perfectly correlated.t . illustrates an important point. If we assume that z1. there was no trade in ﬁnancial assets and specialization in endowments and the allocation was Pareto optimal.t = 1 − α. c2. we can achieve at least two stationary allocations.t+1 . unlike the economy in which there is no trade in ﬁnancial assets. depending on the initial distribution of the claims. In the initial model described earlier.t = z2. U1 (c1. c2.t = βEt U1 (c1. national wealth is equal across countries and agents have perfectly diversiﬁed portfolios.t )p2.21) (2.t+1 + p2. If we introduced trade in equity shares when there is a third good w that is produced by both countries. Fallacy and Fantasy
Let µt denote the Lagrange multiplier.t )q1.t = z2. We commented that we could price ﬁnancial assets even 1 1 if these assets were not traded. c2.t ) = U2 (c1.t )q2. Hence. In particular.t = 1/2 for j = 1. so that zi.t+1 .t .t .t = βEt U1 (c1. then the equilibrium allocation with no trade in ﬁnancial assets can be achieved. International risk sharing can be achieved through ﬂuctuations in relative prices in the current account and by trade in ﬁnancial assets.t+1 ].t+1 ].t+1 )[q2. it does not require the existence of a full set of contingent claims. 2 and i = 1.t+1 )[q1. Agents across countries have identical consumption in this case. The agent maximizes his objective function subject to the constraint.t . c2. c2.t . c2.
j j j j j j j j j j j j
(2. The
. Lucas [152] assumes that agents hold identical j portfolios. The ﬁrst-order conditions are U1 (c1. The presence of nontraded goods or lack of specialization in production of a good can affect how much consumption insurance can be achieved through relative price ﬂuctuations.t = α and 2 2 z1.22)
In equilibrium. If these shocks are perfectly correlated.20) (2.t+1 y2.t+1 + y1.80
j
Business Cycles: Fact. There has been substantial literature on the lack of international portfolio diversiﬁcation and the degree of international consumption risk sharing.

makes it more difﬁcult to determine the optimal degree of consumption risk sharing. combined with the assumption that utility is nonseparable in traded and nontraded goods. optimal behavior would lead to a short position in domestic assets because of a strong positive correlation between the returns to human and physical capital. Devereux. one could ask whether limited risk sharing opportunities among consumers in different countries could help to better reconcile the data with the model. within the context of their model. who show that the data are inconsistent with the implications for consumption for the baseline international RBC model. Nevertheless. that consumption insurance is available through relative price ﬂuctuations and that these price ﬂuctuations are capable of achieving efﬁcient risk sharing. she cannot reject the hypothesis that there is risk sharing. Limited Risk Sharing
As described earlier. Baxter and Jermann [36] argue that the failure of international diversiﬁcation is substantial. Their model incorporates human and physical capital and. and Smith [81] show that preferences that display a nonseparability between consumption and leisure can help to reduce the consumption correlations implied by the model.2. Clearly.
. [25]. As we have noted above. By contrast.International Business Cycles
81
international portfolio diversiﬁcation puzzle is the notion that investors hold too little of their wealth in foreign securities to be consistent with the standard theory of portfolio choice.
4. the existence of nontraded goods.7. A more recent paper by Heathcote and Perri [117] extends the Baxter and Jermann model to include more than one traded good. Lewis [145] also documents that there is insufﬁcient intertemporal risk sharing in consumption. Lewis [145] documents that the nonseparability of utility or the restriction of asset trade alone is not enough to explain the risk sharing that we observe. Gregory. They ﬁnd. whether there exists perfect risk sharing in the data appears to depend on the model speciﬁcation adopted by the researcher. Empirical evidence on international consumption risk sharing is provided in Backus et al. but that when nonseparability and asset trade restrictions are combined. as we have noted above. the conclusion on whether there is sufﬁcient or insufﬁcient risk sharing is very sensitive to model speciﬁcations. We have also shown above that relative price ﬂuctuations can be a substitute for trade in ﬁnancial assets in achieving consumption insurance.

the results for the cross-country correlations are obtained only if the productivity shocks are highly persistent and there are very little spillover effects across countries. Alvarez and Jermann [12]. In their framework. These authors ﬁnd that incomplete asset markets help to reduce the cross-country correlation of consumption. The preferences of the representative consumer in each country are given by
∞
π(s t )U (ci (s t ). They show that the model matches the correlations in the data under a ﬁnancial autarky assumption. In these models. and others on debt-constrained asset markets. Kehoe and Perri [126] consider a model where market incompleteness is obtained endogenously through the introduction of imperfectly enforceable international loans. Kehoe and Perri [126] consider a standard international RBC model with production and capital accumulation. Ai (s t )li (s t )).
t=0 s t
. credit market frictions and restrictions on international capital ﬂows constitute a proposed channel for the propagation of international shocks. Output in each country is produced using capital and labor according to the constant-returns-to-scale production function: F (ki (s t−1 ). where Ai (s t ) is a random shock. Heathcote and Perri [117] examine the implications of a two-country model under alternative assumptions about the ﬁnancial structure. and hours worked remain counterfactually negative.82
Business Cycles: Fact. but the cross-country correlations of output. Moreover. Their analysis involves extending the analysis in Cole and Obstfeld [72] to incorporate an explicit production side. Kollman [136] and Baxter and Crucini [34] examine models where only non-contingent bonds can be traded internationally. investment. given a trade structure. the inability of agents to share risk perfectly and to fully offset the effects of idiosyncratic shocks leads to lower cross-country correlations of consumption and higher cross-country correlations of output. Fallacy and Fantasy
In this vein. 125]. Kocherlakota [135]. li (s t )). a country can incur international indebtedness only to the extent that such indebtedness can be enforced through the threat of exclusion from future intertemporal and interstate trade. This analysis builds on the earlier works of Kehoe and Levine [124.

36. whereas they are positive in the data. which governs the persistence properties of the shocks and the spillovers between them. Second. There is also a borrowing constraint that requires that ¯ ¯ borrowing cannot exceed some ﬁnite bound. the autoregressive coefﬁcients of the shocks for each country captured by the diagonal elements of the A matrix are set at 0. and given by β = 0.85 and the off-diagonal elements equal to 0. the variance of the innovations to the productivity shocks in each country is set at Var( i. (2. the three discrepancies between the model and the data remain.99. and bi (s t ) is the quantity of the bond by consumers in country i. and δ = 0. The coefﬁcients of the A matrix. net exports and investment are much more volatile in the model than they are in the data. 0 < α < 1. the model displays many of the anomalies reported in the literature.15. the consumption correlations are signiﬁcantly higher than the output correlations in the model. A second parameterization allows for high persistence with the diagonal elements equal to 0. α = 0. albeit with some minor improvements in the various statistics. the output and
. Similar to Backus et al.99 and the off-diagonal elements equal to zero. q(s t ) is the period t price of an uncontingent that pays off one unit in period t + 1 regardless of the state. These values are consistent with those assumed by Kollman [136] and Baxter and Crucini [34].
0 < γ < 1.25. According to one parameterization intended to capture substantial persistence.27)
1−σ
. whereas these correlations are the opposite in the data. The ﬁndings from the model echo many of the earlier ﬁndings. Under the complete markets speciﬁcation. where 0 < b < ∞.26)
The parameter values that are considered are standard. In the economy with enforcement constraints. First.36. σ ≥ 0.9 and the off-diagonal elements are set at zero.007 and the correlation of the shocks is set at 0. Preferences in each country are taken to be of the form c γ (1 − l )1−γ U (c. the cross-country correlations of employment and investment are negative in the model.025. In the bond economy. γ = 0.International Business Cycles
85
where wi (s t ) and ri (s t ) denote the wage rate and the rental rate on capital in country i. AL) = k α (Al )1−α . are parameterized in different ways.t ) = 0. and a third one allows for high spillover with the diagonal elements equal to 0. l ) = 1−σ and the production function as F (k.
(2. Third. [25]. σ = 2. b(s t ) ≥ −b.

the volatility of net exports and investment has declined dramatically. leading to very volatile investment and net exports. investment. As discussed earlier. In the literature. they are somewhat dampened. we observe the high cross-country correlations between domestic and foreign consumption. but that the anomalies regarding the cross-correlations of output. Finally. we notice declines in employment and investment. In the foreign country. Fallacy and Fantasy
consumption correlations are both positive and closer to each other. Considering economies with exogenous adjustment costs to investment. although the cross-country correlation of consumption is greater than that of output. Since residents of each country can acquire contingent claims that allow them to smooth consumption across all possible history of shocks. Kehoe and Perri [126] note that the enforcement constraint acts as an endogenous inhibiting factor. In the bond economy. they ﬁnd that the volatilities of investment and net exports are diminished. but because of restrictions on asset trading. Suppose that the social planner tries to implement
. risk sharing across countries implies that consumption increases in the foreign country. This increases the productivity of both capital and labor in the domestic country. the frictionless international RBC model implies that investment ﬂows to the country with the higher productivity shock. and employment remain.86
Business Cycles: Fact. The capital stock in the domestic country increases. Thus. let us consider the impact of a positive productivity shock in the enforcement economy. as domestic residents save more and also as investment from abroad ﬂows there. the cross-country correlations of employment and investment have now switched from being negative to positive. so resources are optimally shifted there. the authors examine the response of the variables in domestic and foreign countries to a positive shock to productivity in the domestic country. The net ﬂow of investment increases the trade deﬁcit in the domestic country. consumption. and third. Second. Consumption increases substantially more in the domestic country because consumption and labor complement each other. Finally. exogenous adjustment costs are typically added to inhibit the ﬂow of investment. The only remaining discrepancy is that net exports are procyclical in the model whereas they are countercyclical in the data. Hence. we see that the differing responses of the domestic versus the foreign country lead to the negative cross-country correlations of employment and investment. the responses are similar to those for the complete markets economy.

increasing the incentives for default. [13] modify the supply side of a two-country model by adding multiple
.They argue that the intranational business cycle is a natural environment for thinking about the interactions between economies when there are no trade frictions and when there are not multiple currencies. this is not possible. the planner must restrict investment to the domestic country to prevent the default option from being exercised. Furthermore. [25] as well as establish the pattern of productivity growth between industries and countries. Hence. Ambler et al. the social planner increases the relative weight on the utility of domestic country residents (recall that the social planner’s weight is the inverse of the marginal utility of consumption).
4.3. and the real exchange rate. and the presence of additional shocks have been developed. As a result. investment.International Business Cycles
87
the complete markets allocation for this economy. but also for the behavior of output. however. In the presence of an enforcement constraint. the production side. We describe a few of these extensions in what follows. OTHER EXTENSIONS
The international RBC literature has sought to reconcile the ﬁndings in the data not only in terms of the cross-country consumption correlations. as the productivity shock dies out. The increased investment ﬂows to the domestic country further increase the value of autarky. the complete markets allocation implies that an increase in output in the domestic country will lead to increases in consumption in both the domestic and foreign countries. the value of autarky diminishes and the relative weight on the home country also falls. Hence. to ensure that domestic country residents consume more than foreign country residents. They then compare these ﬁndings for the international business cycle to those obtained for data between regions within a country — the so-called “intranational business cycle”. the planner also builds up the capital stock in the foreign country to ensure that the risk-sharing arrangement between the domestic and foreign countries is not reneged upon. Hess and Shin [119] re-explore the two international business cycle anomalies emphasized by Backus et al. The high and persistent productivity shock in the domestic country increases the value of autarky. In terms of the behavior of consumption. models that relax assumptions regarding preferences. Over time.

Preferences are assumed to be nonseparable with respect to a composite good. a nontradable good. and leisure. output in the traded and nontraded goods sector of each country is produced using sector-speciﬁc capital and labor which is mobile between sectors. Similar to our discussion above. they argue that what is needed is a source of nation-speciﬁc shocks that shift demand. The model generates a higher crosscountry correlation of output than standard one-sector models. However. they ﬁnd that introducing nontradable goods does not sufﬁce to reduce the cross-country correlation of consumption. or Basu and Kimball [30]). In their model. Fallacy and Fantasy
sectors and trade in intermediate goods. They argue that variable capacity utilization has the potential to account for the co-movement of factor inputs across countries.88
Business Cycles: Fact. It also predicts cross-country correlations of employment and investment that are closer to the data. The authors ﬁnd that the model predicts the correlation between home and foreign output. overstates the cross-country correlation of aggregate consumption. they ﬁnd that the model overstates the negative correlation between the relative price of nontradable to tradable goods and relative consumption of nontraded to traded goods. They introduce taste shocks that affect the utility of traded versus nontraded goods and ﬁnd that this feature brings the data more in line with the implications of the model. Likewise. they argue that introducing nontraded goods may be a way of re-establishing the link between a nation’s output and its spending. They conclude that an international business cycle model driven solely by productivity shocks cannot account for the ﬁndings. We already considered the role of variable factor utilization in reconciling the behavior of procyclical productivity in closed-economy business cycle models (see also Burnside and Eichenbaum [51]. Instead. Stockman and Tesar [200] introduce a nontraded goods sector in each country. Baxter and Farr [35] develop a model with variable capital utilization as a way of accounting for the international correlations. there is no international capital or labor mobility. Hence. The notion is that a positive shock to productivity in the domestic country will tend to reduce the investment
. consisting of the tradable goods of countries 1 and 2. and greatly overstates the cross-country correlation of tradable consumption. They employ preferences that depend on consumption of the tradable goods of countries 1 and 2. Basu [29]. and a nontradable good. and understates the variability of the trade balance.

International trade in assets is made solely with one-period. Output in the domestic country is produced using capital services St and labor Lt as Yt = At St1−α (Xt Lt )α . Kehoe and Perri [126] achieve this through an enforcement constraint which requires that the utility under the constrained allocation exceed the utility that country could obtain under autarky after defaulting on its international debt obligations. Finally. φ > 0. the variance of the innovations is set so that the variance of output in the model exactly matches the volatility
. The parameterization of the capital utilization function and the adjustment cost function are key aspects of the quantitative analysis. many of the puzzles of the international business cycle literature can be addressed successfully by devising a mechanism to limit investment ﬂows across borders. The covariance properties of the technology processes for the domestic and foreign technology shocks are parameterized to be near unit roots with zero spillover effects.
where capital services are the product of the capital stock.International Business Cycles
89
ﬂow across countries by leading to an increase in capital utilization rather than investment. which is consistent with earlier studies. Assuming that the household in each country owns the capital stock and makes real investment decisions. As we described above. δ > 0. and φ < 0. real pure discount bonds which have the price qt = [1 + rt ]−1 . The correlation of the innovations to the technology shocks is set at 0. 0 < α < 1.258. Baxter and Farr [35] assume that preferences are given by the speciﬁcation in equation (2. Kt . where Bt+1 is the quantity of bonds carried over into the next period. Similar functions characterize the foreign country. The capital accumulation process displays both costly utilization of capital and adjustment costs: Kt+1 = [1 − δ(Zt )] + φ(It /Kt )Kt . where δ > 0. Zt : St = Zt Kt . the budget constraint for the representative household is given by Ct + It + qt Bt+1 ≤ Yt + Bt .26). and the utilization rate.

which is desirable. In our earlier discussion. for example. Obstfeld and Rogoff [168] provide a broader view of the empirical puzzles in the international macroeconomics literature. In their review. McCallum [157] or Helliwell [118]. ﬁrst.90
Business Cycles: Fact. introducing variable capital utilization reduces the volatilities of consumption. Fallacy and Fantasy
of the US economy. variable capital utilization takes the place of investment ﬂows across countries. Even with a modest cross-country correlation of the innovations. Obstfeld and Rogoff [168] argue that many of the puzzles in international macroeconomics may just be speciﬁc to the models researchers are using. PUZZLES REVISITED
The puzzles in the international business cycle literature have been the topic of much research (see. Second. capital. and exports. for example. the volatility of the shocks necessary to match output volatility is substantially reduced.
4. They enumerate the following discrepancies between data and existing theory as “puzzles”: • Home bias in trade: A number of authors have documented that intranational trade is typically much greater than international trade. we described the puzzles that arose when trying to match the international RBC model to the data. in a world of integrated capital markets. long averages of saving rates and investment rates tend to be correlated for the OECD countries (see Feldstein and Horioka [89]). Intuitively. One of the key ﬁndings regarding the role of variable capital utilization is that. When a positive productivity shock occurs in one country. capital utilization rates increase. the cross-correlation of hours and investment become positive.4. which is not. and also those of hours and employment. The Feldstein–Horioka puzzle: According to this puzzle. Yet. Speciﬁcally. the survey by Baxter [33]). we would expect capital to ﬂow to regions where the rates of return are highest. which in turn increase employment.
•
. See. The most important impact of variable factor utilization is on the cross-correlations of the factor inputs. this is accompanied by increases in investment and employment in the other country. though they also relate their discussion to the international RBC literature.

we discussed various explanations that account for this puzzle — the presence of human capital as discussed by Baxter and Jermann [36] or nontraded goods. the last two puzzles are pricing puzzles. see Tesar and Werner [203]). In contrast to some of the other contributions that we have discussed. ﬁrst. The purchasing power parity (PPP) puzzle: The PPP puzzle arises from the fact that shocks to the real exchange rate are very persistent. Meese and Rogoff [161] further show that most exchange rate models forecast exchange rates no better than a naive random walk. whereas the other four refer to the behavior of quantities.2. Second. They also note that the pricing puzzles require such features as nominal rigidities of the type that we will consider in the next chapter. the international portfolio diversiﬁcation puzzle. The exchange rate disconnect puzzle: This puzzle captures the notion that there exists a relationship between exchange rates and any macroeconomic variable. the literature that we have described has revealed a variety of promising directions for future research. The international consumption correlation puzzle: We already discussed the ﬁnding that cross-country consumption correlations exceed the crosscountry output correlations in a typical international RBC model. say. In Section 4. Clearly. and does not have the same weight as.
. the analysis of Obstfeld and Rogoff [168] is an attempt to unify the explanations of a related but disparate set of ﬁndings. Obstfeld and Rogoff [168] ﬁnd that adding transport costs goes a long way towards accounting for the ﬁrst ﬁve puzzles. They argue that.
Surprisingly.International Business Cycles
91
•
•
• •
Home bias in equity portfolios: This reﬂects the puzzling preference of stock market investors for home assets (for a recent elaboration. perfect risk sharing across countries implies that countries which experience declines in the relative price of consumption should receive large transfers of goods. whereas the opposite is true in the data. the international consumption correlation puzzle tends to be speciﬁc to a model’s assumptions. Backus and Smith [24] show that in an economy with traded and nontraded goods. as we described above.

This page intentionally left blank
.

output. taken issue with the notion that prices can adjust costlessly to clear markets.1 The behavior of hours and productivity has been a topic of debate. Gali [96] has argued that in a suitably restricted vector autoregression (VAR) including measures of hours.Chapter 5
New Keynesian Models
The New Keynesian approach has gained signiﬁcance in the modern macroeconomics literature. and Kimball [32] also present evidence that hours worked and other variables fall in response to technology improvements in the short run.
93
. Limited participation models also allow alternative mechanisms for the propagation of real and monetary shocks. Basu. productivity. and Evans [66]. This is in contrast to the RBC model. the response of hours to productivity shocks is negative. New Keynesian theories have revived interest in business cycle models that are capable of producing shortrun economic ﬂuctuations based on the types of forces that Keynes had initially postulated. Prototypical New Keynesian models such as that by Rotemberg and Woodford [182] allow for imperfect competition and markups to capture alternative propagation mechanisms in response to technology shocks or shocks to government expenditures. Fernald. from the onset. see Christiano. Their approach involves purging the standard Solow residual of factors that might lead to procyclicality. More recently. Eichenbaum. and other variables to a technology shock have been questioned by empirical results obtained along several different lines. and other variables. which predicts that hours rise on impact to a positive technology shock. The RBC conclusions regarding the response of hours. Critics of the original real business cycle (RBC) approach had.There also exist open-economy versions of the New Keynesian model that have been used for policy analysis
1 For a review and discussion of these models. output. On the one hand. such as variable factor utilization.

price rigidities. THE BASIC MODEL
Consider a simple New Keynesian model with monopolistic competition. see Bowman and Doyle [45]. Suppose that a representative household chooses consumption Ct .5)
ϒt and t denote monetary transfers and proﬁts. In this expression.
(1.4)
is the aggregate price index.
2 For a further discussion.2 We ﬁrst describe a simple New Keynesian framework that can be used to rationalize the observations. and > 1 is the elasticity of substitution among consumption goods. and effort levels Ut to maximize
∞
E0
t=0
βt ln (Ct ) + λm ln
Mt Pt
− H (Nt .94
Business Cycles: Fact. . 1] consumed in period t. and Pt =
1 0 1/(1− ) 1− Pit
di
(1.
5. 1 + σn 1 + σu t (1.2)
for t = 0. 1.
.1)
subject to
1 0
Pit Cit di + Mt = Wt Nt + Vt Ut + Mt−1 + ϒt +
t
(1.. and variable labor effort due to Gali [96]. money holdings Mt .3)
where Cit is the quantity of good i ∈ [0.1. The functional form for H (Nt . 2. and then discuss the empirical ﬁndings in more detail. Ut ) is given by H (Nt . Ut ) = λn λu Nt1+σn + U 1+σu . The price of good i is given by Pit . Ut )
(1. hours worked Nt . . respectively. Ct is a composite consumption good deﬁned as Ct =
1 0
Cit
( −1)/
/( −1)
di
. respectively. Wt and Vt denote the nominal prices of an hour of work and effort. Fallacy and Fantasy
and for understanding the role of monetary shocks. .

where µt denotes the Lagrange multiplier on the period-by-period budget constraint. We can solve for µt from the ﬁrst-order condition corresponding to the consumption choice as µt = 1/(Pt Ct ). Substituting for this variable and simplifying yields Cit = Pit Pt
−

where A = ((1 − θ)/θ(λn /λu ))α(1−θ)/(1+σu ) . Substituting this result into the equation for the production function yields Yt = AZt Nt ,
ϕ

(1.26)

where ϕ = αθ + (1 + σn )α(1 − θ)/(1 + σu ). Using the price-setting rule in (1.22) together with (1.12) and the expression for equilibrium consumption

27) (1. we can show that pt = ξt−1 − (1 − γ)ηt−1 . given a constant money supply and predetermined prices. producing the same output will require less labor input. Hence. Hence. 1 1−γ ξt − ηt . employment. which corresponds to the situation of short-run increasing returns to labor. and productivity.
.98
Business Cycles: Fact. A positive technology shock deﬁned by ηt > 0 has a permanent positive one-for-one impact on output and productivity and a permanent negative impact on the price level if γ < 1. Fallacy and Fantasy
and output derived above. The impact of ξt on labor productivity is also transitory. and xt depend only on the current ξt . with a positive shock to technology. but the sign depends on whether ϕ < (>)1. when there is no accommodating response in the money supply to real shocks. ϕ
where x = y − n is the log of labor productivity. an increase in ξt causes output and employment to go up for one period and then to revert back to their initial values. A positive monetary shock deﬁned by ξt > 0 has a temporary impact on output.27)–(1. Finally. In such a case. However. though with a one-period lag. nt . We note that measured labor productivity responds positively whenever ϕ > 1. demand remains unchanged so that ﬁrms will be able to meet demand by producing an unchanged level of output.29)
xt = 1 −
+ (1 − γ) 1 −
1 ηt−1 . yt = nt = ξt + γηt + (1 − γ)ηt−1 . that is.30) can be used to describe the impact of monetary versus technology shocks. More interestingly. This result can be best understood by considering the case of γ = 0.30) (1. and hence a decline in employment will occur. real balances remain unchanged in the face of a positive technology shock.28) (1. the price level responds one-for-one to an increase in ξt . ϕ ϕ 1 ϕ ξt + 1−γ + γ ηt ϕ (1. The conditions in (1. This can be observed by noting that the levels of yt . a positive technology shock has a negative short-run impact on employment.

the long-run impact of a given shock. EMPIRICAL EVIDENCE
Gali [96] estimates a structural VAR (SVAR) and identiﬁes technology shocks as the only shocks that are allowed to have a permanent effect on average labor productivity.
where z and m denote the sequences of technology and non-technology t t shocks. Furthermore. Gali [96] ﬁnds that variables that have no permanent effects on employment (and which are referred to as demand shocks) explain a substantial fraction of the variation in both employment and output.2. implies that E ( t t ) = I . Eichenbaum. which can be expressed as the restriction C 12 (1) = 0. technology shocks explain only a small fraction of employment and output ﬂuctuations. i. together with a normalization. respectively. and Vigfusson [68] suggest that the standard RBC results hold if per capita hours are measured in log-levels as opposed to differences. they argue that if the simulated data from a simple RBC model are subjected to a SVAR-based test.
. The SVAR model interprets the behavior of (log) hours nt and (log) productivity xt in terms of two types of exogenous disturbances — technology and non-technology shocks — which are orthogonal to each other and whose impact is propagated through time based on some unspeciﬁed mechanisms as xt nt = C 11 (L) C 12 (L) C 21 (L) C 22 (L)
z t m t
= C (L) t . the difference-stationary version of the model implies that the response
3The value of the matrix C (L) evaluated at L = 1 gives the long-run multipliers for the model. Kehoe.3 Gali [96] estimates this model using postwar US data. The identifying assumption is that only technology shocks have a permanent effect on productivity. and McGrattan [63] have challenged the ﬁndings derived from so-called SVARs. Surprisingly. This is similar to the approach in Blanchard and Quah [43] for identifying demand versus supply shocks using long-run restrictions on estimated VARs. Christiano. By contrast.e. they argue that the difference speciﬁcation for aggregate hours is a priori misspeciﬁed because all RBC models imply that per capita hours is a stationary variable. Second.. he ﬁnds that alternative measures of labor input decline in response to a positive technology shock while GDP adjusts only gradually to its long-run level. The orthogonality assumption.New Keynesian Models
99
5. Chari. First.

on the one hand. Basu et al. if hours worked is considered to be stationary in levels. and various prices and interest rates to changes in the puriﬁed Solow residual. non-residential investment. generated substantial controversy and. Following Gali [96] and Gali and Rabanal [98]. Unlike the SVAR approach.100
Business Cycles: Fact. non-constant returns to scale. they advance price rigidity as the major reason for these deviations from the RBC predictions in the short run. leads to an erroneous conclusion of the negative response of hours worked to a productivity shock. variable factor utilization. and sectoral re-allocation and aggregation effects. in their view. They ﬁnd that purging the standard Solow residual of these effects eliminates the phenomenon of “procyclical productivity”.
. These ﬁndings have. They use both standard regression analysis and simple bivariate VARs for this purpose. even though the data underlying this test are drawn from a standard RBC model which implies the opposite response! Despite the controversies surrounding the SVAR approach. [32] present evidence that support the SVAR ﬁndings by generating a modiﬁed Solow residual that accounts for imperfect competition. cast further doubt on the ability of the RBC model driven by technology shocks to provide a convincing explanation of economic ﬂuctuations for the major developed countries. hours worked. They also examine the response of a key set of variables such as output. Fallacy and Fantasy
of hours to a shock to productivity is negative as in Gali and Rabanal [98]. This misspeciﬁcation. durables and residential investment. employment. non-durables and services. on the other. utilization. the evidence obtained from this approach is robust to long-run identifying assumptions or to the inclusion of new variables in the estimated dynamic system. Their ﬁndings corroborate the ﬁndings from the SVAR approach regarding the negative response of hours to technology improvements in the short run. then they argue that the conﬁdence bands for the impulse response functions from the SVAR are so wide that the procedure is uninformative about the question at hand. They also uncover further evidence for the negative response of non-residential investment to such shocks. They trace the source of the difference to the failure to include a sufﬁcient number of lags in the estimated VAR speciﬁcations so as to adequately capture the behavior of hours implied by the underlying theoretical model. However.

they have been examining the efﬁcacy of RBC-type models in accounting for these facts. On the other hand. Ratfai and Benczur [175. Argentina. and income and exports are typically highly volatile. and Rebelo [77]. consumption varies more than output. for example. More recently. Neves. They argue that the large shortfalls in GDP suffered by Argentina in the 1980s can be explained in a simple growth model framework using the observed measure of productivity or the Solow residual. Aguiar and Gopinath
101
. the real business cycle (RBC) agenda has been increasingly applied in emerging market contexts. Arellano and Mendoza [17]). but their analysis is based on the one-sector optimal growth model. for example. Business cycles in emerging markets exhibit different characteristics compared to those in developed economies. Emerging market business cycles also display a different set of stylized facts relative to developed economies. researchers have begun generating business cycle facts for such economies (see. For instance.Chapter 6
Business Cycles in Emerging Market Economies
In recent years. The question then arises whether a small open economy-type RBC model can account for both developed and emerging market business cycles. Kydland and Zaragaza [144] also study the behavior of an emerging market economy. On the one hand. 176]). the trade balance is strongly countercyclical. Earlier RBC models for small open economies were developed by Mendoza [162] and Correia. namely. The recent literature on the “Sudden Stop” phenomenon has emphasized the large reversals in current accounts and the incidence of capital outﬂows that have become identiﬁed with many recent emerging market economies’ experience (see.

as well as the correlations of the latter three with output. is 0. there are also overidentifying conditions which can be used to test the model’s implications. These ﬁndings have been challenged by Garcia-Cicco et al.41 for Canada depending on the speciﬁcation for preferences. an emerging market economy version estimated using quarterly data for Mexico and a developed country version estimated for Canada between 1980 and 2003. This ratio captures the importance of shocks to trend productivity. Fallacy and Fantasy
6.4 for Mexico. consumption. The authors argue that differences in the magnitude of shocks to trend productivity can account for some of the salient features of business cycles in emerging market economies. and 2. φ). ρz . they show that a positive transitory shock to productivity reduces consumption in anticipation of lower income in the future. The moments that they use for estimation include the standard deviations of log (ﬁltered) income.2. In particular. By contrast. these results can also explain why consumption is more volatile than income in economies where shocks to productivity growth are more important than transitory shocks.25 or 0. Since the number of parameters is less than the number of moment conditions.106
Business Cycles: Fact. who argue that the results of Aguiar and Gopinath [1] are due to their use of
. as is the capital adjustment parameter φ. DO SHOCKS TO TREND PRODUCTIVITY EXPLAIN BUSINESS CYCLES IN EMERGING MARKET ECONOMIES?
Aguiar and Gopinath [1] consider two versions of their model. investment. ρg . This leads to a large trade deﬁcit which tends to persist for a considerable period of time (16 quarters). Thus. Furthermore. σz . [101]. They also consider the mean and standard deviation of (unﬁltered) income growth as well as the autocorrelations of (ﬁltered) income and (unﬁltered) income growth. net exports-to-GDP. By contrast. The most noteworthy ﬁnding that emerges from the estimation is that the ratio of shock standard deviations. σg . The parameters to be estimated are given by (µg . the shock to trend productivity can generate consumption booms that tend to appear side-byside with current account deﬁcits in emerging market economies. This yields 11 moment conditions.5 or 5. σg /σz . the autocorrelations of transitory shocks are roughly similar. a positive shock to trend productivity causes consumption to respond more than output in the expectation that income will be higher in the future.

Second. they show that output ﬂuctuations in the post-World War II period are as large as those in the pre-World War II period. [101] use annual data for Argentina over the period 1913–2005. They consider a model that is very similar to the one in Aguiar and Gopinath [1]. investment growth. the permanent shock is estimated to be more volatile and persistent than the transitory shock. In particular. and estimate the parameters of the stochastic processes for the permanent and transitory shocks using 16 moment conditions. and. and the trade balance-to-output ratio. and ﬁnd that the consumption-smoothing motive in response to a transitory shock dominates the anticipatory effect of consumption to a permanent productivity shock. They determine this empirically.and second-order autocorrelations of output growth. In particular. this choice of sample period does not seem out of line. Whereas the ﬁrst result suggests that the estimated model does not emphasize the role of permanent shocks sufﬁciently. consumption growth will be more volatile than output growth if permanent shocks are more important than transitory shocks. As we described above. As we described earlier. Garcia-Cicco et al. However. and less volatile than output growth if the opposite is true. in terms of the parameter estimates. Given the evidence that business cycles have moderated during this period. they consider the variances and ﬁrst. The authors also ﬁnd that the trade balance-to-output ratio is estimated to be around four times as volatile as output growth. First. consumption growth in the estimated model is calculated to be less volatile than output growth. in contrast to what is observed in the data. and the unconditional mean of output growth. much of the business cycle literature for developed countries has concentrated on the post-World War II period. which is not signiﬁcantly different from zero.Business Cycles in Emerging Market Economies
107
a short sample to estimate low-frequency movements in productivity. By contrast. [101] argue that a similar ﬁnding is not true for an emerging market economy such as Argentina. Garcia-Cicco et al. [101] ﬁnd that the overidentifying restrictions of the model are rejected. investment growth. the second result suggests that it overemphasizes them. whereas in the data these quantities are roughly equal. Third. this is not the case for the autoregressive coefﬁcient for the transitory shock. and the trade balance-to-output ratio. The standard deviations of the shocks and the autoregressive parameter for the permanent shock are estimated precisely. Garcia-Cicco et al. unlike Aguiar and Gopinath [1]. the correlations of output growth with consumption growth. investment growth is found
. consumption growth.

as households perceive the income increase to be permanent. the trade balance-to-output ratio displays a near random walk behavior even though the estimated autocorrelation function in the data is downward-sloping. In particular. In particular. the behavior of the trade balance-to-output ratio again follows a near random walk because in this case. By contrast. Garcia-Cicco et al. Hence. If the shocks are permanent. with a constant world interest rate. they argue that some of their ﬁndings that we described above point to the importance of shocks that are different from productivity shocks. regardless of whether shocks to productivity are permanent or transitory. if the shocks are transitory.
. suggesting in this case that neither source of shocks is sufﬁciently volatile. The authors also show that the random walk behavior of the trade balance-to-output ratio remains. consumption increases in response to an innovation to output in roughly the same magnitude as output. they also argue that their results are derived under the joint hypothesis of business cycles driven by productivity shocks and the propagation mechanisms of the standard RBC approach. Fallacy and Fantasy
to be insufﬁciently volatile.108
Business Cycles: Fact. their ﬁndings cannot be used to disentangle which of these factors is responsible for the failure of the model to replicate the observations. consumption in the model follows a near random walk even though output and investment become stationary variables. [101] conclude that a pure RBC model driven by permanent and/or transitory exogenous shifts in productivity does not provide an adequate explanation of business cycles in emerging markets. However. Hence. The authors show that the estimated model also fails to replicate the autocorrelations of output growth and the trade balance-tooutput ratio. the trade balance is unaffected and the trade balance-tooutput ratio inherits the behavior of output.

We also discussed variations of the model that could be used to account for speciﬁc correlations in the data. This model was initially proposed by Sargent and Sims [185] for describing cyclical phenomena. One of the most popular approaches has been based on the dynamic factor model. In her original contribution. which seeks to describe the joint cyclical behavior of a key set of time series in terms of a lowdimensional vector of unobservable factors and a set of idiosyncratic shocks. The recent business cycle literature has witnessed the use of a variety of techniques regarding the empirics of business cycles. who used the generalized method of moments (GMM) approach (see Hansen [113]) to match a selected set of unconditional ﬁrst and second moments implied by their model. Amongst the most prominent of these assumptions is the assumption of perfect price ﬂexibility. Their approach may be viewed as an extension of the standard RBC approach. An alternative approach was proposed by Christiano and Eichenbaum [65]. Watson [209] extended this approach to derive measures of ﬁt for an underlying economic model. which assesses the adequacy of the model based on the behavior of the relative variability and co-movement of
109
. In this chapter.Chapter 7
Matching the Model to the Data
In the previous chapters. Yet one of the most important aspects of the debate regarding the RBC approach lies in the use of calibration as a way of matching the model to the data. we will study alternative approaches for matching the implications of a theoretical model with the data. we described some criticisms leveled against the assumptions of the real business cycle (RBC) framework. and also provide an overview of the estimation versus calibration debate. Altug [5] estimated an unobservable index model for a key set of aggregate series based on a modiﬁed version of the Kydland and Prescott model [141] using maximum likelihood.

we require that ˜ E (f˜t νi. DYNAMIC FACTOR ANALYSIS
Dynamic factor analysis seeks to describe the joint cyclical behavior of a key set of time series in terms of a low-dimensional vector of unobservable factors and a set of idiosyncratic shocks that are mutually uncorrelated and uncorrelated with the factors. . .1)
˜ ˜ where {H (s)}∞ s=−∞ is a sequence of (n × k)-dimensional matrices.1. According to this model. Canova [56. Under these assumptions.
(1.110
Business Cycles: Fact.t−s ) = 0 for all ν ˜ i. the variances and autocovariances of the observed series {wt } can be decomposed in terms of the variances and ˜ autocovariances of a low-dimensional set of unobserved common factors and
. n for i = j. let {wt }∞ denote an n-dimensional mean zero. E (f˜t f˜t−s ) = 0 for t = s and E (˜ i. covariation among the elements of wt can ˜ arise because they are functions of the same common factor or because they are functions of different factors which are themselves correlated at different leads and lags. and νt is an n × 1 vector of idiosyncratic shocks ˜ that are mutually uncorrelated and uncorrelated with common factors. that is.3)
Both the common factors and the idiosyncratic factors may be serially correlated. j.t νi.t νj.2) (1.t ) = 0 ν ˜ for i = 1. Finally. covariance stationary stochastic ˜ t=0 process used to describe observations on the (possibly de-trended) values of a set of variables. 57] discusses alternative approaches for conducting statistical inference in calibrated models. t = s. More precisely. ft is a k × 1 vector of common factors.
7. Bayesian estimation of the so-called dynamic stochastic general equilibrium (DSGE) models provides an alternative to incorporating prior beliefs about various parameters that formalize some of the practices in the calibration approach.t ) = 0 E (˜ i. To describe how to formulate unobservable index models. . A k-factor unobservable index model for wt is given by ˜
∞
wt = ˜
s=−∞
˜ ˜ H (s)f˜t−s + νt . . (1. Fallacy and Fantasy
a small set of moments.

˜ ˜ where H (ω) denotes the Fourier transform of H (s). Unlike the unrestricted factor model which can be estimated frequency by frequency.d.i. Hence. The restricted factor model makes use of the cross-equation restrictions across the linear decision rules implied by the original model. The use of the dynamic factor model in business cycle analysis dates back to the work of Sargent and Sims [185]. the dynamic factor model provides decomposition at each frequency that is analogous to the decomposition of variance in the conventional factor model. Sargent [184] also employs the device of augmenting a singular model with additional idiosyncratic shocks. Also. As these authors observe. He discusses a classical measurement error case. who derives an unobservable index model for a key set of aggregate series by augmenting the approximate linear decision rules for a modiﬁed version of the Kydland and Prescott [141] model with i. The dynamic factor model can be estimated and its restrictions tested across alternative frequencies using a frequency domain approach to time series analysis. as in Altug [5]. and the idiosyncratic shocks are interpreted as i. Altug also uses this representation to estimate the model using maximum likelihood (ML) estimation in the frequency domain. measurement errors or idiosyncratic components not captured by the underlying RBC model. Another well-known application of this approach is due to Altug [5].d. The unrestricted version of the dynamic factor model does not place restrictions on ˜ the matrices H (s).i. it is not possible to identify ˜ the common factors with different types of shocks to the economy. which describe how the common factors affect the behavior of the elements of wt at all leads and lags. this model must be estimated jointly across
. error terms. as well as a case with orthogonal prediction errors.112
∞
Business Cycles: Fact. dynamic factor analysis may be linked to the notion of a “reference cycle” underlying the methodology of Burns and Mitchell [50] and the empirical business cycle literature they conducted at the National Bureau of Economic Research. The common factor is identiﬁed as the innovation to the technology shock. Fallacy and Fantasy
∞
=
s=−∞
˜ H (s) exp (−iωs)
∞
Rw (u) exp (−iωu)
u=−∞
×
z=−∞
˜ H (z) exp (−iωz) + Sν (ω)
˜ ˜ = H (ω)Sw (ω)H (ω) + Sν (ω).

The ACGF for yt is similarly denoted Ay (z). Also. model. and Ax summarizes the unconditional second-moment properties of this model. The question at hand is whether the data generated by the model are able to reproduce the behavior of the observed series. Watson’s analysis does not depend on assuming that the unobserved factors and the idiosyncratic shocks are uncorrelated. when the restrictions of the underlying model are imposed. not to describe a statistical model that can be used to conduct statistical tests. In the data. She ﬁnds that the hypothesis of a single unobservable factor cannot be rejected at conventional signiﬁcance levels for describing the joint time series behavior of the variables. deﬁne the n × 1 vector of
2 For further discussion of maximum likelihood estimation in the frequency domain. To describe Watson’s approach. his approach involves choosing the correlation properties of the error process between the actual data and the underlying model such that its variance is as small as possible. investment in structures. the joint process for the data and the error is introduced to motivate goodness-of-ﬁt measures.
7.
. the model cannot explain the cyclical variation of the observed variables. However.
denotes the stochastic process generated by some underlying Here.Matching the Model to the Data
113
all frequencies because the underlying economic model constrains the dynamic behavior of the different series as well as speciﬁes the nature of the unobserved factor.1. see Hansen and
{xt }∞ t=0
Sargent [114].2 Altug [5] initially estimates an unrestricted dynamic factor model for the level of per capita hours and the differences in per capita values of durable goods consumption. Measures of Fit for Calibrated Models
Watson [209] extended the approach in Altug [5] and Sargent [184] to derive measures of ﬁt for an underlying economic model. the vector of variables yt corresponds to the empirical counterpart of xt . and aggregate output.1. investment in equipment. For this purpose. Deﬁne the autocovariance generating function (ACGF) for xt by
∞
Ax (z) =
r=−∞
E (xt xt−r )z r . let xt denote an n × 1 vector of covariance stationary random variables. Unlike the approach adopted by Altug and Sargent. Instead.

the difference between Ay (z) and A error.3 To continue. If. However. and Ax (z) from the model.6) (1. Ay (z) = Ax (z). Fallacy and Fantasy
errors ut that are required to reconcile the ACGF implied by the model with that of the data. This implies a version of a classical errors-in-variables approach. xt . 4This is similar to the interpretation provided by Altug [5]. the assumption of pure measurement error which is uncorrelated with the true variable is not appropriate. Then.5)
where Axy (z) is the joint ACGF for xt and yt . in addition.7)
ˆ a given model ﬁts that data. it is the size of the sampling error that is used to judge whether
.5 In the ﬁrst case. let Ay (z) denote the population autocovariance function and Ay (z) ˆy (z) is ascribed to sampling denote its estimated counterpart. which implies that the ACGF for ut can be expressed as Au (z) = Ay (z) + Ax (z) − Axy (z) + Ayx (z). In this case. The problem is to choose σxy to minimize the variance of ut subject to the constraint that the covariance matrix of xt and yt remains positive deﬁnite. notice that Ay (z) can be determined from the data. Nevertheless. σxy
(1. 5 We omit the third case that Watson considers because it requires results for the Cramer representation of stationary time series. yt .114
Business Cycles: Fact. Hence. the assumption regarding Axy (z) is that it is zero. In the standard dynamic factor model. let us consider two cases.t. in Watson’s framework. Axy (z) = 0. then the sampling error in estimating Ay (z) from actual data can also be used to determine how different Ay (z) is from Ax (z).
3 In a standard goodness-of-ﬁt approach. This bound is calculated by choosing Axy (z) to minimize the variance of ut subject to the constraint that the implied joint ACGF for xt and yt is positive deﬁnite. and the size of the sampling error can be determined from the data-generating process for yt . deﬁne ut by the relation ut = yt − xt . |σxy | ≤ σx σy . or |σxy | ≤ σx σy :
2 2 2 min σu = σy + σx − 2σxy s. the error term ut is the approximation error in describing the observed data with the underlying economic model. a lower bound may be deduced for the variance of ut without imposing any restrictions on Axy (z).4 However. To calculate Au (z). and ut are assumed to be serially uncorrelated scalar random variables. since Axy (z) is unknown. (1. some additional assumptions are needed to operationalize this approach. To illustrate this approach.

this result corresponds to the one in the scalar case. yt ) is positive semideﬁnite. One convenient transformation is the trace of u . An alternative approach is to minimize the weighted sum of the variances as tr(W u ). tr( u ) = n i=1 ij. Now suppose that xt and yt are serially uncorrelated random vectors with covariance matrices x and y . we cannot minimize the variance of ut directly. the joint covariance matrix of (xt . He log-linearizes
. which minimizes the weighted sum of the variances of the elements of ut subject to the constraint that this matrix is positive semideﬁnite. where W is an n × n matrix. Cx is the n × k matrix square root of x as x = Cx Cx and Cy is the n × n matrix square root of y . yt ) is singular.u denotes the i. given the properties of the U and V matrices. xt can be represented as xt = yt . Speciﬁcally. For k = 1. σu = (σy − σx )2 at the minimum. is given by
xy
= Cx VUCy . V is a k × k orthonormal matrix (with VV = Ik ).9)
In this expression. The problem in this case is to choose xy to minimize tr(W u ) subject to the constraint that the covariance matrix of (xt . In this case. Furthermore. The covariance matrix of ut is given by u = y + x − xy + yx . Watson [209] provides a solution for the case in which x has rank k ≤ n so that the number of variables is typically less than the number of shocks. Instead. we consider a transformation that allows us to determine the size of ut . As a consequence. (1. and Rebelo [131.Matching the Model to the Data
115
It is easy to see that the solution for this problem is to set σxy = σx σy . as in the scalar case.10) where = Cx UVCy−1 . and S is a k × k diagonal matrix.8)
where γ = σx /σy .
(1. An implication of this result is that. The matrices U and V are deﬁned from the singular value decomposition USV of Cy WCx such that U is a n × k orthogonal matrix (with U U = Ik ). Watson [209] uses this methodology to generate goodness-of-ﬁt measures for a standard RBC model with a stochastic trend in technology.u . Proposition 1 in Watson [209] demonstrates that the unique matrix xy . 132]. As a 2 consequence. xt and yt are perfectly correlated so that xt = γyt . he considers the model in King. where ij. Plosser. (1. j element of .

investment.1.
6 A comparison of the spectra implied by the model and those generated by actual data also ﬁgured in early versions of Altug [5]. and investment occur at frequencies corresponding to the business cycle periodicities of 6–32 quarters. His approach allows for a decomposition by frequency of the variance in each observed series. Reichlin. the economy-wide shocks can be identiﬁed using structural vector autoregression (SVAR) techniques. They examine the behavior of four-digit industrial output and productivity for the US economy for the period 1958–1986 and ﬁnd evidence in favor of at least two economy-wide shocks. However. and the error in reconciling the model with the data.116
Business Cycles: Fact. and the unobserved factor model can be estimated by using equation-by-equation ordinary least squares (OLS). They also describe how to derive impulse response functions for time series models which have reduced rank. and the number of hours worked.2. the variance explained by the model. that is. Fallacy and Fantasy
the ﬁrst-order conditions to obtain the solution for log-differences of output.
7. consumption. their results also indicate that sector-speciﬁc shocks are needed to explain the variance of the series. Giannone. Other Applications
Forni and Reichlin [93] use the dynamic factor model to describe business cycle dynamics for large cross-sections. and Sala [103] show how more general classes of equilibrium business cycle models can be cast in terms of the dynamic factor representation. Based on a law of large numbers argument. they show that the number of common factors can be determined using the method of principal components. both having a long-run effect on sectoral output.6 He ﬁnds that the biggest differences between the spectra for output. Watson’s analysis shows that focusing only on a small subset of moments implied by the model can be misleading because the RBC model is unable to reproduce the typical spectral shape of economic time series. The model also implies that the number of hours worked is stationary whereas there is considerable power at the low frequencies for this series in the data.
. suggesting that the stochastic trend properties of the model do not hold in the data. consumption. see also Altug [4]. models for which the number of exogenous shocks is less than the number of series.

For example. Their estimation strategy is based on a subset of the ﬁrst and second moments implied by their model. 1 = (δ.Matching the Model to the Data
117
7. g . Christiano and Eichenbaum [65] derive a solution for the social planner’s problem by implementing a quadratic approximation to the original nonlinear problem around the deterministic steady states. θ. let 1 denote a vector of parameters determining preferences. Their approach may be viewed as an extension of the standard RBC approach. technology. GMM ESTIMATION APPROACHES
Other papers have employed nonlinear estimation and inference techniques to match equilibrium business models with the data. and the share of capital in the neoclassical ¯ production function.
(2. To describe how their approach is implemented. ρ.11)
. and the exogenous stochastic processes. which assesses the adequacy of the model based on the behavior of the relative variability and co-movement of a small set of time series. σλ ) . As in the standard RBC approach. σµ . θ.2. γ. More generally. the deterministic steady states are derived for the transformed variables. λ. the elements of restrictions E H1t (
1)
satisfy the unconditional moment = 0.
Proceeding in this way. Some of the parameters included in 1 may be the depreciation rate of capital. the depreciation rate δ is set to reproduce the average depreciation on capital as E δ− 1− kt+1 it − kt kt = 0. the parameters in 1 are estimated using simple ﬁrst-moment restrictions implied by the model. Likewise. Christiano and Eichenbaum [65] consider the hours-productivity puzzle and use the generalized method of moments (GMM) approach (see Hansen [113]) to match a selected set of unconditional ﬁrst and second moments implied by their model. δ. Since there is a stochastic trend in this economy arising from the nature of the technology shock process. the share of capital satisﬁes the Euler equation E β−1 θ yt+1 kt+1 +1−δ
1
ct ct+1
= 0.
given data on gross investment it and the capital stock kt+1 .

the authors show that the statistic J = F ( ˆ T ) Var[F ( ˆ T )]−1 F ( ˆ T ) (2. Using data on private consumption. once we quantify the uncertainty in model predictions arising from uncertainty about model parameter values. Pancrazi.3. Since then. calibrated or otherwise. As we described earlier.
7. our view of what the data is [sic] telling us is affected in a ﬁrst-order way. Even if
7 See Kydland and Prescott [142. Kydland and Prescott [141] initiated this debate in the modern business cycle literature.Matching the Model to the Data
119
In practice. Christiano and Eichenbaum [65] estimate the parameters of the model using GMM and examine the various unconditional second moments implied by the model. THE CALIBRATION VERSUS ESTIMATION DEBATE
The crux of the recent business cycle debate centers on how a given theoretical model should be matched with the data.
. and Uribe [101] employ this approach in their analysis of business cycles in emerging markets. we will deal with a variety of criticisms aimed directly at the calibration approach and some suggested alternatives to it. Eichenbaum [82] asked whether RBC analysis. constituted “wisdom or whimsy”. the containing T observations. He took issue with the calibration approach by showing that the model’s implications for the variance of output relative to its value in the data are heavily dependent on assumed values for the underlying parameters for the technology process. Letting test statistic for the second-moment restrictions is based on the distribution of F ( ˆ T ) under the null hypothesis. in fact. government expenditures. aggregate hours. there have been ongoing discussions on both sides of the issue. Using this distribution.21)
is asymptotically distributed as a χ2 random variable with two degrees of freedom. For example. In a highly suggestive analysis. In his words:
Indeed. investment. the GMM approach has been used in other recent applications.7 In this section. there is sampling error in estimating from a ﬁnite data set ˆ T denote the estimated value of . Aguiar and Gopinath [1] and Garcia-Cicco. 143] for a further discussion and defense of their approach. and average productivity.

whereas government shocks evolve as a persistent AR(1) process. The Dynamics of Output
Cogley and Nason [70] examine the dynamics of output as a way of determining the efﬁcacy of the RBC model in describing the data.3. which includes shocks to productivity and government consumption as well as the indivisible labor feature of Hansen [112] and Rogerson [179]. To estimate impulse response functions from the data.
Eichenbaum [82] also presented evidence to show that the performance of the standard model as well as modiﬁcations that allow for labor hoarding. they use the two-shock version of the RBC model proposed by Christiano and Eichenbaum [65].1. What the data are actually telling us is that.120
Business Cycles: Fact. Such a break accounts for the slowdown in productivity growth in the US that occurred in the late 1960s. They assume that the technology shocks are difference-stationary. The answer could be 70% as Kydland and Prescott [142] claim. They consider a two-variable VAR with output and hours worked (or consumption). its power spectrum. Fallacy and Fantasy
we do not perturb the standard theory and even if we implement existing formulations of that theory on the standard postwar sample period and even if we use the stationary inducing transformation of the data that has become standard in RBC studies — even then the strong conclusions which mark this literature are unwarranted. the authors use the SVAR technique developed by Blanchard and Quah [43]. Since the single technology shock model is singular. while technology shocks almost certainly play some role in generating the business cycle. The striking results in Cogley and Nason [70] show that the autocorrelation function
. for example. but the data contain almost no evidence against either the view that the answer is really 5% or that the answer is really 200%. deteriorate considerably when a break in the sample is allowed for. and identify the technology shock under the assumption that it has a permanent effect on output. they consider the autocorrelation function for output. and impulse response functions in response to shocks. The autocorrelation function from the model is generated by simulating the model 1000 times. As part of their diagnostics.
7. They compare the autocorrelation function and the impulse response function in the data with those generated by the model using generalized Q statistics. there is simply an enormous amount of uncertainty of just what percent of aggregate ﬂuctuations they actually do account for.

I ) denote the joint density for the data and the parameters. Cogley and Nason [70] also argue that the model displays weak propagation mechanisms which arise from intertemporal substitution. conditional on the function f and the parameters β. Also. Deﬁne expectations of the functions of the simulated data
8These results are. conditional on the information set available to the researcher I and the function f . the model has some success in matching the estimated response functions in the data for the permanent component of GDP. investment. See Cogley and Nason [69]. f ) denote the density for the parameters β. Denote the predictive density p(Xt |I . as is the power spectrum. and the spectrum for output displays signiﬁcant power at the business cycle frequencies of 2. His approach involves constructing prior distributions for the parameters used in calibrated models based on existing estimates in the literature or other a priori information available to the ¯ researcher. β) denote the process for these variables generated by a speciﬁc economic theory or model as a function of exogenous and predetermined variables Zt and the parameters β. let Xt = f (Zt . complementary to those regarding the role of ﬁltering on the cyclical
properties generated from a standard RBC model. Let G(Xt |f . Calibration as Estimation
Canova [56. and interest rates. To describe his approach.3. let π(β|I . β) denote the density of the vector Xt . say. In particular. in some sense. the autocorrelations of output at lags 1 and 2 are signiﬁcant and positive. 57] proposes an alternative approach for providing statistical inference in calibrated models.33–7 years per cycle. capital accumulation.8 By contrast.
7.Matching the Model to the Data
121
for output implied by the model is nearly ﬂat. β|f . but it is unable to match the impulse response function for the transitory component.2.
. and adjustment costs. let Xt denote a vector stochastic process with a known distribution that describes the evolution of a set of observed variables. In terms of the impulse response functions. The test statistic also shows that the implications of the model are rejected at signiﬁcance levels of 1% or less. f ) = H (Xt . the data display a hump-shaped response to transitory shocks whereas the model implies a much smaller monotonic decay. GDP. f )d β. and let H (Xt . β|I .

• Repeat the above two steps N times. • Construct the frequency distribution of µ(xt ) and other statistics of interest that can be used to evaluate the performance of the model. suppose that we are interested in evaluating Pr(ν(Xt ) ∈ D). model speciﬁcations. One of the key aspects of Canova’s approach is the selection of a density π that summarizes information about existing parameter estimates in an efﬁcient manner. the approach that Canova advocates for conducting statistical inference in calibrated models is as follows: • Select a density π(β|I . • For each drawing of β and zt . In general. f )dXt H (Xt . I ) = = µ(Xt )p(Xt |I .22)
This approach allows for a probability statement regarding the behavior of various moments implied by the model. β|I . one could count estimates of elements of β obtained from alternative studies and smooth the resulting histogram to obtain the density π(β|I . where ν(Xt ) is some moment of interest and D is a bounded set. The calibration approach involves putting a point mass on a particular value of β. Notice also that the function f is typically unknown and must be obtained using some approximation method. • Draw vectors β from π(β|I . Fallacy and Fantasy
(denoted by µ(Xt )) under the predictive distribution p(Xt |I .122
Business Cycles: Fact. (3. deﬁne µ(Xt ) = 1 if ν(Xt ) ∈ D and zero otherwise. f )d βdXt . Then. the model can be simulated repeatedly for different values of β and Zt to compute sample paths for Xt . Canova [56] describes how to account for such an approximation error when computing the moments of the simulated data. While the densities H and p are unknown. More precisely. f ) and zt from k(Zt ). then one can use a uniform distribution. Simulation exercises conducted after a subset of the parameters has been estimated (using GMM. for
. f ) for the unknown parameters. For example. This information may be derived from alternative data sets. f ) by E (µ(Xt )|f . f ). β) or its approximation. If such information is hard to obtain. or estimation techniques. generate {xt }T and compute µ(xt ) using t=1 the model xt = f (zt . given the information I available to the researcher and a density k(Zt ) for the exogenous processes.

3. Alternatively. increasing returns. Ashley and Patterson [22] developed a test to test for deviation from linear stochastic processes. Nonlinearity in Macroeconomic Time Series
Another criticism of the RBC model literature is that it has typically been concerned with examining the ﬁrst. Neftci [166] was among the ﬁrst to demonstrate that unemployment ﬂuctuations were asymmetrical along the business cycle. they do not ﬁnd evidence of nonlinearity in the Solow residuals measured after allowing for increasing returns to scale. Ashley.3. variable capacity utilization. and industrial production could be described as nonlinear stochastic processes. One can argue that the approach described above is a global sensitivity exercise that also allows for an evaluation of the model based on the probabilities attached to events that the researcher is interested in. they note that observed measures of output and factor inputs such as the capital stock and hours worked display marked nonlinear behavior. Yet there is a new literature that shows that macroeconomic time series may exhibit marked nonlinear behavior. They conclude that the nonlinearity must lie in the transmission mechanism. Using an array of diagnostic tests. and Patterson [9] examine the implied behavior of output. and argued that any reasonable macroeconomic model should display some form of nonlinear dynamics (see Potter [172] for a review).
7. and time-varying markups.
. Such nonlinearities may take the form of conditional heteroscedasticity such as ARCH or GARCH effects. or variable capacity utilization. and the underlying productivity shocks using a simple production function framework. unemployment. They found strong evidence of nonlinearity in industrial production. either in the form of nonlinear stochastic dynamics or deterministic chaos. GMM. say. Speciﬁcally. markups. and while they could not ﬁnd evidence of chaotic behavior. there may exist asymmetries in various economic variables.Matching the Model to the Data
123
example) involve putting a point mass on β∗ estimated according to a particular method. Brock and Sayers [47] tested real macroeconomic variables displayed by deterministic chaotic dynamics. In our discussion in Chapter 3. they nevertheless showed that postwar employment. In a novel analysis. Altug. the factor inputs.and second-moment properties of aggregate economic variables for the purpose of matching a model to the data. we discussed the role of alternative factors that could give rise to endogenous changes in productivity such as labor hoarding.

but they are unable to reproduce nonlinearities in these time series.e. The ﬁrst model selected by the SNP is a VAR(3) with conditional heteroscedasticity and a Hermite
. excess kurtosis). The objective function is distributed as a X 2 statistic. Japan. In the ﬁrst step.. and shows that nonlinearities such as skewness. the economic model is simulated for a given set of parameters. and the UK. it also allows for periods of high volatility followed by low volatility (conditional heteroscedasticity). kurtosis. The SNP approach nests standard VARs. The statistical models are ﬂexible enough to allow for increasing time dependence in the mean of the process (captured through a VAR). asymmetric business cycles (i. Fallacy and Fantasy
Valderrama [207] examines the statistical behavior of national income and product account aggregates for the US and a set of OECD countries including France. skewness). In the second step. Mexico. However. He poses this as a “canonical” challenge to the RBC approach. He argues that standard general equilibrium models are able to replicate the ﬁrst. and for nonnormal disturbances. as in the GMM approach. Valderrama [207] considers a simple Brock–Mirman-type growth model with GHH preferences (see Greenwood. The last feature is achieved by taking a transformation of the normal density using Hermite polynomials (see Gallant and Tauchen [100]). A comparison is made between the statistical parameter estimates in the SNP step and the statistical parameters obtained using the simulated data and the same statistical model. for conditional heteroscedasticity (ARCH. which is a two-step procedure. Valderrama [207] selects three statistical models to describe the properties of the data. and conditional heteroscedasticity are common for many of these aggregates. a seminonparametric (SNP) model is estimated to characterize the statistical properties of the data.e.and second-moment properties of such variables. Hercowitz. The SNP approach uses a standard VAR to model the conditional mean and an ARCH-GARCH structure to model the conditional variance. This ensures that the nonlinearities in the underlying model are not eliminated through a linearization procedure. Italy. and excess volatility (i. GARCH).124
Business Cycles: Fact. and Huffman [105]) and adjustment costs in investment.. Then. The approach to matching the model to the data is through the efﬁcient method of moments (EMM) developed by Gallant and Tauchen [99]. the candidate parameters of the simulated model are adjusted until the simulations of the economic model have statistical properties similar to those of the data. and it allows for a non-Gaussian error term. A value iteration approach is used as the solution procedure.

this parameter is estimated to be around 10. φ. these results suggest that such features as ﬁnancial frictions or irreversibility in investment are required to match the nonlinearities of consumption and investment. but not the nonlinearity in consumption.Matching the Model to the Data
125
polynomial of degree 4. Combined with the ﬁndings of Altug et al. The parameters of the underlying economic model are chosen based on standard calibration exercises and also to match the three statistical models using a simulated method of moments approach. When the statistical model is forced to be a linear VAR(3). its estimates are around 3. and arrive at results that indicate little or no signiﬁcant impact of irreversibility or lumpiness on aggregate investment dynamics. Surprisingly. The ﬁrst of these is a linear VAR(3) so that the statistical procedure is similar to the RBC approach of matching the impulse responses from a standard VAR with those generated from the underlying economic model. implying that the irreversibility feature is not important for matching the model to the data. This is similar to some earlier results in the literature. The linear VAR(3) model also implies a much lower volatility for the consumption and investment series than the other two models. By contrast. the other two statistical models allow for these features and hence do not require such a high adjustment cost parameter. The second model is a VAR(3) with ARCH(1) errors. which allows for nonlinearity in terms of the ARCH effects but continues to assume that the errors are Gaussian. The intuition for this result stems from the fact that a high adjustment cost parameter helps to smooth the implied behavior of investment and to reduce conditional volatility or kurtosis in the investment series. Valderrama [207] ﬁnds that the irreversibility constraint does not bind during the solution of the model. Two other models are selected for the purpose of describing the data. it can capture the nonlinearity in investment. Valderrama ﬁnds that the RBC model is not successful at generating the conditional variance of investment. whereas in the other two models. [9] who show that the nonlinearities in the observed series must lie in the propagation mechanism. Veracierto [208] argues that the reason why irreversibility has an impact on aggregate investment in various
. Veracierto [208] and Thomas [205] compare model economies with irreversible investment (in the case of Veracierto) or lumpy investment (in the case of Thomas) with the actual economy and with a baseline model of ﬂexible investment. Valderrama [207] ﬁnds that the biggest difference among the three statistical models is in terms of the adjustment cost parameter.

the aggregate impact of irreversible investment disappears in a general equilibrium framework. ﬁnancial accelerator models of investment ﬁnd that ﬁnancially constrained ﬁrms’ investment responds three times as strongly to a monetary expansion than that of ﬁrms which are not constrained (see Bernanke. changing the way in which the labor market is modeled and introducing wage contracts in the RBC framework may lead to more pronounced quantity adjustments. for example. for otherwise they would not be identiﬁed. Moreover. these microeconomic nonlinearities must matter at the aggregate level. we suggest that the reason why irreversibility may not matter in a model with ﬂexible prices is that the introduction of imperfect competition with optimal price-setting behavior on the part of ﬁrms or. Thus. the reason why aggregate investment displays lower variability in the presence of irreversibility when ﬁrms are subject to idiosyncratic shocks. Veracierto [208] argues that when the size of sectoral shocks is consistent with that observed in the data. is due to their assumption of a very large variance for these idiosyncratic shocks. namely. At the same time. Both Veracierto [208] and Thomas [205] assume that labor is perfectly mobile across plants. More generally. and Gilchrist [41]). However. Gertler. Fallacy and Fantasy
multi-sector growth models considered in the literature (see. as Valderrama [207] states. Similarly. Given the importance of nonlinearity in macroeconomic time series documented in a number of studies. As Caballero [53] emphasizes: “An important point to note is that since only aggregate data were used. this is a puzzling ﬁnding. These arguments suggest that nonlinearities are another
. one may consider the impact of another type of aggregate shock. irreversibility may play a more important role when ﬁnancial constraints at the plant level and the household level are taken into account. both Veracierto and Thomas ﬁnd that irreversibility or lumpiness at the plant level does not affect aggregate investment signiﬁcantly once general equilibrium effects such as endogenous price adjustments are taken into account. for example. and may be responsible for softening the impact of the inﬂexibility faced by plants in terms of their capital adjustments. This latter feature may compensate for the rigidity faced by plants given that capital and labor are substitutable according to the Cobb–Douglas speciﬁcation of technology.” Yet. Coleman [73] or Ramey and Shapiro [174]) is due to their assumption of unrealistically large sectoral shocks. monetary shocks. alternatively. as in Bertola and Caballero [42].126
Business Cycles: Fact. our analysis of the simple RBC model with ﬂexible prices and wages suggests that it cannot successfully match many features of the data.

Watson’s [209] insights were partly due to Altug’s initial analysis based on the spectra implied by the model and those in the data. and a potential direction for improving the model’s ﬁt. highlighting problems in ﬁnding appropriate
9 See Altug [4]. for example. 17).3. but in what sense does this make the model a better interpretation of reality? King also discusses the shortcomings of this approach. King claims to be on the quantitative theory side of the debate.
. For some time. and argues against the estimation and comparison of a “heavily restricted linear time series model (for example.
7. Yet. One can also examine the claim that a less restrictive approach to estimation and model evaluation following the approach in Christiano and Eichenbaum [65].9 In the absence of the initial frequency-domain estimation based on the factor representation. Many have argued that this was to be expected (see Hartley. one could argue that the GMM approach in Christiano and Eichenbaum [65] involves choosing a subset of moments of an equally restrictive nonlinear model. King [128] advocates such an approach as an alternative to calibration that does not face the problems of “testing highly restrictive linear models”. an RBC model with some or all of its parameters estimated) to an unrestricted time series model. we note that Altug’s [4] results did not lead to support for the model. it is unlikely that RBC models would be subjected to the types of diagnostic tests proposed by Watson [209]. may be preferable. The factor representation underlying Altug’s analysis has also been resuscitated by Giannone et al.” Returning to the inception of the estimation versus calibration debate. The Debate Reconsidered
King [128] discusses the arguments on different sides of the debate under the heading “Quantitative Theory and Econometrics”. the outcome of this procedure will be known in advance of the test: the probability that the model is true is zero for stochastically singular models and nearly zero for all other models of interest. [103] in a VAR framework. Yet. p. If the simple model is counterfactual.4. and Salyer [116].Matching the Model to the Data
127
area where the assumptions of the simple RBC model fail to hold. in what sense does adding another simple feature to such a contrived model “resolve” a very speciﬁc empirical puzzle? Adding another shock may “loosen” the behavior of the model. Hoover.

128
Business Cycles: Fact. the estimation versus calibration debate has led to a panoply of research that has extended the use of these models in a variety of directions. Fallacy and Fantasy
instruments and in the appropriate selection of moment conditions for model evaluation. The initial criticisms of
. Perhaps initially the RBC theorists were concerned that the estimation versus calibration debate would discredit the use of well-speciﬁed dynamic general equilibrium models for the purpose of capturing the quantitative behavior of economic variables. This interpretation precludes the notion that quantitative theorizing can take the place of formal econometric methods. and the New Keynesian challenge has shown that the model fails in generating the observed negative response of hours to productivity improvements. far from this occurring. one could argue that the approach in Christiano and Eichenbaum [65] is too restrictive. In fact. In this sense. It is also worth noting that many of the implications of the standard RBC model have not withstood the scrutiny conducted under a variety of approaches. examining a small set of moments does not necessarily lead to a less restrictive approach because the moments under consideration typically incorporate all the implications of the underlying theoretical model. The simple propagation mechanisms inherent in the model have been deemed incapable of reproducing business cycle dynamics of key variables such as output. Gregory and Smith [108] have also argued that the smallsample properties of the estimators obtained with the approach advocated by Christiano and Eichenbaum [65] may be far from reasonable if the calibrated parameters do not consistently estimate the true values. These arguments show that there is no clear-cut dichotomy between these approaches. As Geweke [102] notes. Another criticism of the RBC approach is that it fails to capture nonlinearities in key macroeconomic time series. Conversely. One interpretation of quantitative theorizing is that it helps researchers understand the workings of highly nonlinear dynamic stochastic models and gain some intuition about different model features. These developments show that the contribution of the more formal econometric techniques need not be dismissed as cursorily as is sometimes the case. The estimation and testing of a highly restrictive linear or linearized model may yield many insights regarding the failure of a model as much as examining the performance of a model based on a small subset of moments. examining a broader set of moments may make more sense because this yields more information about the underlying behavior of the model.

For example. Above all. there is no banking sector and no speciﬁc role for money and credit in the monetary transmission mechanism. Another extension of the original approach has been the development of applications that are useful for policy analysis. and Scott [123] for policy analysis of a small open economy. as all agents have complete knowledge of the economy and complete understanding of shocks when they hit. a topic to which we turn next. There is also an interest-rate-setting rule that deﬁnes
. As we discuss in the next chapter. The aim of this model is to reproduce the inertial behavior of inﬂation and persistence in real quantities. In sum. The model assumes a representative household and a symmetric equilibrium for ﬁrms. One can ask whether this class of models overcomes some of the criticisms leveled against the original RBC approach.
Yet this model has a core set of 26 equations! As another example. ﬁrms are assumed to borrow working capital to ﬁnance their wage bill. the original RBC approach has even instigated a new generation of Keynesian models that offer features such as credit accelerators. Finally. To capture the ﬁrst feature. There are no market frictions and no locational speciﬁcity.4. especially in the short run. DSGE MODELING
Another offshoot of the original RBC debate is the development of dynamic stochastic general equilibrium (DSGE) models that can be used for policy analysis. and animal spirits. and a variety of techniques have been developed for the purpose of matching the model to the data. sunspot equilibria. the model incorporates wage and price contracts following the approach in Calvo [54]. consider the model proposed by Christiano. The model also incorporates a variety of “real” frictions such as habit persistence in consumption. the model recently proposed by Kapetanios. and Evans [67] for the analysis of the effects of monetary policy shocks on real and nominal variables. no ﬁxed costs or discontinuities. There are no market frictions and distortions.
7. and variable capital utilization. The recent class of models developed for this purpose has vastly more features than any simple RBC model. adjustment costs in investment. Eichenbaum. Pagan. for example. markets are assumed to clear at all times. Consider. the model contains assumptions that are almost guaranteed to be violated by the data.Matching the Model to the Data
129
the RBC approach have led to a wide set of extensions of the original RBC approach. According to them:
[This] model is stark in its assumptions.

but homogeneous in their consumption and asset holdings. Demers. the impulse responses to a given monetary shock. By contrast. Christiano et al. [67] pursue a limited information estimation strategy by estimating a subset of the parameters of the model to match the impulse responses of eight key macroeconomic variables to a monetary policy shock with those implied from an identiﬁed VAR using the so-called method of indirect inference. and Altug [79]. Demers. Furthermore. [67] assume that monetary policy shocks are drawn from a given and known distribution. see Altug. Yet. with the latter result derived from the existence of state-contingent securities for consumption. This raises
10 For a recent example of the method of indirect inference applied to a model of the EU economy. For applications. Christiano et al.130
Business Cycles: Fact. It is also very much in the spirit of the original Kydland–Prescott approach [141] in that the goal is to match a set of observed characteristics — in this case. the model assumes a continuum of households which are heterogeneous in their wage and the hours that they work.11 Christiano et al. see Meenagh. the irreversible investment model studied by Demers [78] and others has been shown to generate a time-varying adjustment cost that varies in response to changes in objective and subjective forms of risk and uncertainty. provide some evidence on the role of price and wage rigidities.
. but in a highly restrictive environment. Christiano et al. 11 For a review and discussion. one could also question the implications of the model should there be changes in the distribution of the exogenous processes. However.10 The authors’ analysis is. in many ways. the assumption of a constant adjustment cost parameter would fail to hold. and Demers [10. there are shortcomings in their approach. Fallacy and Fantasy
monetary policy. [67] derive much of their evidence regarding the types of real rigidities on the results of calibration-type exercises of the current generation of macroeconomic models such as habit persistence or adjustment costs. the adjustment cost model has been criticized on the grounds that it implies a constant cost of adjustment which does not vary with economic conditions. For example. Minford. an exploratory analysis of the role of nominal and real rigidities in propagating shocks. see also Demers. 11]. implying that the propagation mechanisms would vary with changes in the distribution of exogenous variables facing agents. In such cases. For example. and Wickens [160]. estimated models of adjustment costs have been shown to imply implausible degrees of costs of adjustment.

Schorfhiede [186] and An and Schorfhiede [15]). a variety of other papers have implemented maximum likelihood estimation of dynamic equilibrium models. in fact. suppose that the state space representation for the model’s solution consists of the following: • a transition equation St = g (St−1 . There are a variety of approaches for solving nonlinear dynamic stochastic models. they augment the approximate linear laws of motion with additional shocks. which is obtained from the prior distribution and the likelihood function. More recent applications include Canova [60] and Ireland [121]. for example. and information. where St is a vector of state variables that describe the evolution of the model over time. Rogerson. and Wright [159] consider an equilibrium business cycle model with household production and distortionary taxation. and θ is a vector of parameters characterizing preferences. First. To illustrate this approach. the likelihood function for the observations must be formed. McGrattan. amongst others. the use of Bayesian methods and. which is typically a high-dimensional object. However. this approach ﬁrst provides a link with the calibration approach by allowing the researcher to use information from microeconomic studies or macroeconometric estimates. Similar to Canova [56]. in particular. As in the analysis of Altug [5]. Third. The Bayesian estimation of DSGE models involves several steps. Second. the posterior distribution for the parameters. the method presupposes that a solution can be found for the underlying economic model of interest. the production technology. The recent research on DSGE models has also employed Bayesian analysis as a way of incorporating prior uncertainty about the parameters of interest (see. vt . Following the initial contribution of Altug [5]. examining the posterior distribution calculated as the prior distribution times the likelihood function is often more straightforward computationally than maximizing the likelihood function. vt is a vector of innovations. Judd [122] provides a textbook treatment of many currently used solution methods. must be examined. Second. structural or invariant to interventions.Matching the Model to the Data
131
the issue of whether DSGE models are. and
. θ). they employ time domain methods by making use of the Kalman ﬁlter with a linear law of motion for the state variables and a linear measurement equation for a key set of observables.

θ). θ)p(St |y t−1 . This state space representation can be used to derive the likelihood function of the observables and to obtain the posterior distribution for the parameters conditional on the observations. which shows t=1 the conditional distribution of the states given the observations. p(y T |θ)π(θ)d θ (4. yT ) at the parameter values θ as
T
p(y . θ) = p(y1 |θ)
t=2
T
p(yt |y t−1 . this sequence is computed by making use of the relation12 p(St+1 |y t . the posterior distribution can be expressed as π(θ|y T ) = p(y T |θ)π(θ) .23)
The remaining task is to compute the sequence {p(St |y t−1 . . θ)}T . we can also compute the probabilities p(St |St−1 . θ)dS0
t=2
p(Yt |St . Following his approach. wt . . Likewise. θ) and p(Yt |St . θ). θ). θ)dSt
12This is known as the Chapman–Kolmogorov equation. the state space representation allows us to compute the likelihood of the observations y T = (y1 . θ) = p(St+1 |St . The excellent survey by Fernandez-Villaverde [90] describes in detail how to implement Bayesian estimation of DSGE models. among others). Given the probability density functions for the shocks fv ( · ) and fw ( · ). θ)
T
=
p(S1 |S0 . θ). . wt . Fallacy and Fantasy
• a measurement equation Yt = h(St . . Notice that Yt = h(g (St−1 . θ)dSt .
Given a prior distribution π(θ) for the parameters θ. we can compute p(Yt |St−1 . Hence.132
Business Cycles: Fact. where Yt are the observables and wt are the shocks to the observables (which may take the form of measurement errors.
. θ)p(St |y t . vt . θ).

Matching the Model to the Data
133
and an application of Bayes’ theorem as p(St |y t . Their estimates imply more price stickiness than the estimates of Christiano et al. Smets and Wouters [194] argue that their approach offers several important advances. θ) = p(yt |St . then the Kalman ﬁlter is used to obtain the likelihood function. Otherwise. They extend the approach in Christiano et al. if the transition and measurement equations are linear and if the shocks are normally distributed. [67]. They consider the behavior of seven key macroeconomic time series in the euro area — real GDP. First. [67] to estimate the parameters of a DSGE model with real and nominal frictions for the euro area using Bayesian methods. the estimated DSGE model performs as well as standard and Bayesian VARs. [67]. For example. consumption. a method known as the particle ﬁlter13 is used to evaluate the likelihood function. First. p(yt |St . a labor supply shock. on the whole. Fourth. a temporary contractionary monetary shock has a temporary negative effect on output and inﬂation. their ﬁndings attribute the largest role
13 See Schorfhiede [186] or Fernandez-Villaverde [90]. the real wage. investment. however. based on Bayesian model evaluation criteria. In contrast to the approach in Christiano et al. θ)p(St |y t−1 .
. These include a productivity shock. Smets and Wouters [194. the GDP deﬂator. θ)p(St |y t−1 . 195] provide two important examples of this approach. and the nominal short-term interest rate — by minimizing the posterior distribution model’s parameters based on a linearized state space representation for the model. they ﬁnd that the estimates of the structural parameters of the model are plausible and. they consider a full set of structural shocks. Second. employment.23) involves a few remaining steps. θ)dSt
Finding the posterior distribution of the parameters displayed in (4. even if we can evaluate the posterior distribution. similar to those for the US economy. θ) . they ﬁnd that. and a government consumption shock plus three “cost-push” shocks. a shock to the household’s discount factor. Third. However. exploring it for different values of θ typically involves the use of sampling through Markov chain Monte Carlo (MCMC) methods. an investment-speciﬁc shock. they argue that the effects of the different shocks on the variables of interest are consistent with existing evidence that does not rely on a DSGE framework.

134

Business Cycles: Fact, Fallacy and Fantasy

to labor supply and monetary shocks in accounting for the variation in output in the euro area. One problem with this approach, as we discussed above, is that there tends to be a circularity in building a model to match ﬁndings, say, from a standard VAR, which may not be robust themselves. In his survey written in 1992, Fair [86] expressed his disappointment with both the RBC and New Keynesian research agendas in the following way: “The RBC literature is interested in testing in only a very limited way, and the New Keynesian literature is not econometric enough to even talk about serious testing.” In the last 20 years or so, the New Keynesian agenda has progressed much closer towards gaining an explicitly econometric focus. Yet, it is difﬁcult to say how much headway has been made in understanding the propagation mechanisms underlying business cycle ﬂuctuations. Simple models with features ranging from home production to government consumption shocks to nonseparable preferences have been examined to understand their inner workings, as have the roles of nominal rigidities and price-setting behavior. While these model features have proved useful for accounting for some stylized facts, they have often continued to retain counterfactual implications. For example, Cogley and Nason [70] show that the home production framework produces a negative autocorrelation for output. As discussed earlier, many of the model features, such as symmetric and constant adjustment costs in investment, are essentially ad hoc and are at variance with empirical evidence at the micro level. The recent DSGE models include a large set of shocks that have little economic meaning. By its very construction, the DSGE framework cannot easily move into a setup that relaxes the representative consumer assumption, be this through observed and unobserved forms of heterogeneity or the presence of uninsurable risk. Yet, even under the complete markets assumption, Altug and Miller [7, 8] show that exogenous and endogenous forms of heterogeneity matter for the behavior of individual allocations. Browning, Hansen, and Heckman [48] provide a comprehensive discussion regarding the role of heterogeneity in general equilibrium modeling, and raise cautionary points about linking the current class of dynamic equilibrium models that dominate macroeconomics with microeconomic models and evidence. Could the entire general equilibrium macroeconomic quantitative theorizing approach — despite the use of much bigger models and more powerful computational tools — be a detour, as Fair [86] seems to indicate?

Matching the Model to the Data

135

Should economists build intuition from the workings of well-articulated economic models at the same time as they use more purely econometric approaches for quantitative policy analysis? Kapetanios et al. [123] note that the current class of fully articulated dynamic equilibrium models that are increasingly being used by policy-makers such as central banks help to provide restrictions on a reduced form, a practice very much in the spirit of the Cowles Commission approach. Thus, the exercise is not necessarily to recover structure in the form of the parameters of preferences and technology, as argued by Lucas [150] in his famous critique of econometric policy evaluation. More tellingly, can economists hope to recover “true structure” using macroeconomic data in the presence of observed and unobserved forms of heterogeneity? Or, alternatively, when subjective beliefs by economic agents in a changing stochastic environment constitute “hidden structure”, as Martin Weitzman [210] claims? The notion that structural models may not indeed be structural has begun to be discussed in the literature. Fernandez-Villaverde and Rubio-Ramirez [91], for example, discuss the stability over time of estimated parameters in DSGE models, and show that the parameters characterizing the pricing behavior of households and ﬁrms in Calvo-type pricing functions are correlated with inﬂation. Canova and Sala [61] investigate identiﬁability issues in DSGE models and their consequences for parameter estimation and model evaluation when the objective function measures the distance between estimated and model impulse responses. They show that observational equivalence as well as partial and weak identiﬁcation problems are widespread, lead to biased estimates and unreliable t-statistics, and may induce investigators to select false models. Canova [59] also examines the issue of identiﬁability of DSGE models, and argues that researchers should be careful in interpreting the diagnostics obtained from the estimation of structural models and make use of additional data in the form of micro data or data from other countries to augment the information obtained by formal estimation.

This page intentionally left blank

For one. However. For example. Farmer [88] generates a model of self-fulﬁlling equilibria which arises from market failure associated with the labor market. I have omitted a discussion of multi-sector models in generating business cycle behavior. I have offered a perspective based on my own perceptions and mindsets. In this book. even the experience of reviewing some of this literature shows that it is lively.Chapter 8
Future Areas for Research
The literature on business cycles is vast. it may be that shocks are concentrated in a particular sector and are propagated to the rest of the economy from that sector. Speciﬁcally. the wage is then set by a
1 See. I have not provided a discussion of models with selffulﬁlling expectations and multiple equilibria. Nevertheless. Farmer [87] provides an eloquent enunciation of this approach. More generally.
137
. and contrasts it with the standard real business cycle (RBC) approach. but always thought-provoking. There are many other topics of interest that we could have discussed regarding business cycles. Many have argued that such models are more in the spirit of true Keynesianism because they stress the role of “animal spirits”. multi-sector models may also have shortcomings because they typically imply that overall expansions in the economy are associated with contractions in one sector.1 In such cases. one needs to consider multi-sector models since the one-sector growth model does not allow for a consideration of these issues. Many recent works assume that there is a search technology such that ﬁrms and workers are randomly matched. he considers the role of a search externality and a lemons problem in the market for search inputs. for example. In a recent analysis. Benhabib and Farmer [38]. at times contentious.

we may conclude that the debate surrounding their development will not cease to arouse interest or to generate future avenues for research. Undoubtedly.138
Business Cycles: Fact. he assumes that ﬁrms offer the same wage in advance. This feature of the model allows for the role of “conﬁdence”. which seeks to alleviate demand-induced shortfalls in output through large ﬁscal stimuli. see Pierrard. One could argue that this type of model deserves much more attention given the global ﬁnancial crisis that originated from the subprime housing market in the US in 2007. new techniques.3 The above considerations show that business cycles will continue to be a topic of discussion among economists for many years to come. see Cooley and Quadrini [76]. Farmer [88] drops the assumption that wages are determined as a result of a Nash bargaining process. This leads to a model with a continuum of steady-state equilibria. For a recent application in the context of a standard RBC model. This leads to a demand-constrained equilibrium.2 In contrast to this literature.
2This approach owes its origins to the analysis in Mortensen and Pissarides [165]. and new controversies regarding both. Fallacy and Fantasy
Nash bargaining process.
. The policy implications of such a model differ from those of the standard textbook Keynesian model. this discussion will be accompanied by the development of new models. 3 For a recent analysis of models with ﬁnancial frictions in a dynamic general equilibrium setting. all ﬁrms earn zero proﬁts but not all equilibria have the same welfare properties. which is transmitted to real variables through investors’ beliefs about the stock market. By contrast. This book has also omitted a discussion of models with credit and collateral constraints. Instead. However. Farmer [88] concludes his analysis by commenting on the potential efﬁcacy of the Obama ﬁscal stimulus package(s) in light of the ﬁndings from his model. irrespective of what the new models or techniques will look like. de Walque. In these equilibria. the model with a search externality emphasizes the role of conﬁdence in restoring full-employment output. and Rouabah [171].